Determining properties of row spaces of two column-equivalent matrices











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Given matrix $$A = begin{pmatrix} 1 & r & 1 & 1 \ 1 & s & 1 & 2 \ 1 & t & 1 & 3 \ 1&0&0&1end{pmatrix}$$ and let $B$ be the matrix obtained from $A$ by exchanging the $2^{nd}$ and $4^{th}$ columns, i.e. $$B = begin{pmatrix} 1&1&1&r\1&2&1&s\1&3&1&t\1&1&0&0 end{pmatrix}$$



(i) Justify whether the column space of $A$ the same as the column space of $B$.



(ii)Justify whether the row spaces of $A$ and $B$ have the same dimension.



(iii) Let $C$ by an invertible 4-by-4 matrix. Determine a basis for the row space of the 4-by-8 matrix $$D = begin{bmatrix} mathbf{C} & mathbf{C}end{bmatrix}$$



For part (i), this is my attempt:



I note that $B^T$ is the matrix obtained from $A^T$ by swapping the $2^{nd}$ and $4^{th}$ rows of $A^T$, thus $B^T$ is row equivalent to $A^T$ since elementary row operations preserve the row space $-$ $A$ has the same column space as $B$.



For part (ii), I base my attempt on my answer from (i). $A$ and $B$ have the same column space, so $A$ and $B$ must have the same dimension of column space. Since $$dimRan(A) = dimRan(A^T)$$ and $$dimRan(B) = dimRan(A)$$ we get that $dimRan(B^T) = dimRan(A^T)$ i.e. the row spaces of $A$ and $B$ indeed have the same dimension.



For part (iii), I make use of the fact that $$rref(C) = begin{pmatrix} 1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1end{pmatrix}$$ so $$rref(D) = begin{bmatrix} 1&0&0&0&1&0&0&0\0&1&0&0&0&1&0&0\0&0&1&0&0&0&1&0\0&0&0&1&0&0&0&1end{bmatrix}$$



Thus we can take a basis for the row space of $D$ to be $${begin{pmatrix}1\0\0\0\1\0\0\0end{pmatrix},begin{pmatrix}0\1\0\0\0\1\0\0end{pmatrix},begin{pmatrix}0\0\1\0\0\0\1\0end{pmatrix},begin{pmatrix}0\0\0\1\0\0\0\1end{pmatrix}}$$



I'm pretty sure my answer for (i) is correct, but I'd just like to verify and see if there would be another way to explain it, but I'm unsure about part (ii) as intuitively, I would expect the row spaces of $A$ and $B$ to have different dimensions since $A$ is not row-equivalent to $B$.



My approach for part (iii) seems a little too naive and I feel like there's more to the question than meets the eye.



Help would be appreciated!










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    Given matrix $$A = begin{pmatrix} 1 & r & 1 & 1 \ 1 & s & 1 & 2 \ 1 & t & 1 & 3 \ 1&0&0&1end{pmatrix}$$ and let $B$ be the matrix obtained from $A$ by exchanging the $2^{nd}$ and $4^{th}$ columns, i.e. $$B = begin{pmatrix} 1&1&1&r\1&2&1&s\1&3&1&t\1&1&0&0 end{pmatrix}$$



    (i) Justify whether the column space of $A$ the same as the column space of $B$.



    (ii)Justify whether the row spaces of $A$ and $B$ have the same dimension.



    (iii) Let $C$ by an invertible 4-by-4 matrix. Determine a basis for the row space of the 4-by-8 matrix $$D = begin{bmatrix} mathbf{C} & mathbf{C}end{bmatrix}$$



    For part (i), this is my attempt:



    I note that $B^T$ is the matrix obtained from $A^T$ by swapping the $2^{nd}$ and $4^{th}$ rows of $A^T$, thus $B^T$ is row equivalent to $A^T$ since elementary row operations preserve the row space $-$ $A$ has the same column space as $B$.



    For part (ii), I base my attempt on my answer from (i). $A$ and $B$ have the same column space, so $A$ and $B$ must have the same dimension of column space. Since $$dimRan(A) = dimRan(A^T)$$ and $$dimRan(B) = dimRan(A)$$ we get that $dimRan(B^T) = dimRan(A^T)$ i.e. the row spaces of $A$ and $B$ indeed have the same dimension.



    For part (iii), I make use of the fact that $$rref(C) = begin{pmatrix} 1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1end{pmatrix}$$ so $$rref(D) = begin{bmatrix} 1&0&0&0&1&0&0&0\0&1&0&0&0&1&0&0\0&0&1&0&0&0&1&0\0&0&0&1&0&0&0&1end{bmatrix}$$



    Thus we can take a basis for the row space of $D$ to be $${begin{pmatrix}1\0\0\0\1\0\0\0end{pmatrix},begin{pmatrix}0\1\0\0\0\1\0\0end{pmatrix},begin{pmatrix}0\0\1\0\0\0\1\0end{pmatrix},begin{pmatrix}0\0\0\1\0\0\0\1end{pmatrix}}$$



    I'm pretty sure my answer for (i) is correct, but I'd just like to verify and see if there would be another way to explain it, but I'm unsure about part (ii) as intuitively, I would expect the row spaces of $A$ and $B$ to have different dimensions since $A$ is not row-equivalent to $B$.



    My approach for part (iii) seems a little too naive and I feel like there's more to the question than meets the eye.



    Help would be appreciated!










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      Given matrix $$A = begin{pmatrix} 1 & r & 1 & 1 \ 1 & s & 1 & 2 \ 1 & t & 1 & 3 \ 1&0&0&1end{pmatrix}$$ and let $B$ be the matrix obtained from $A$ by exchanging the $2^{nd}$ and $4^{th}$ columns, i.e. $$B = begin{pmatrix} 1&1&1&r\1&2&1&s\1&3&1&t\1&1&0&0 end{pmatrix}$$



      (i) Justify whether the column space of $A$ the same as the column space of $B$.



      (ii)Justify whether the row spaces of $A$ and $B$ have the same dimension.



      (iii) Let $C$ by an invertible 4-by-4 matrix. Determine a basis for the row space of the 4-by-8 matrix $$D = begin{bmatrix} mathbf{C} & mathbf{C}end{bmatrix}$$



      For part (i), this is my attempt:



      I note that $B^T$ is the matrix obtained from $A^T$ by swapping the $2^{nd}$ and $4^{th}$ rows of $A^T$, thus $B^T$ is row equivalent to $A^T$ since elementary row operations preserve the row space $-$ $A$ has the same column space as $B$.



      For part (ii), I base my attempt on my answer from (i). $A$ and $B$ have the same column space, so $A$ and $B$ must have the same dimension of column space. Since $$dimRan(A) = dimRan(A^T)$$ and $$dimRan(B) = dimRan(A)$$ we get that $dimRan(B^T) = dimRan(A^T)$ i.e. the row spaces of $A$ and $B$ indeed have the same dimension.



      For part (iii), I make use of the fact that $$rref(C) = begin{pmatrix} 1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1end{pmatrix}$$ so $$rref(D) = begin{bmatrix} 1&0&0&0&1&0&0&0\0&1&0&0&0&1&0&0\0&0&1&0&0&0&1&0\0&0&0&1&0&0&0&1end{bmatrix}$$



      Thus we can take a basis for the row space of $D$ to be $${begin{pmatrix}1\0\0\0\1\0\0\0end{pmatrix},begin{pmatrix}0\1\0\0\0\1\0\0end{pmatrix},begin{pmatrix}0\0\1\0\0\0\1\0end{pmatrix},begin{pmatrix}0\0\0\1\0\0\0\1end{pmatrix}}$$



      I'm pretty sure my answer for (i) is correct, but I'd just like to verify and see if there would be another way to explain it, but I'm unsure about part (ii) as intuitively, I would expect the row spaces of $A$ and $B$ to have different dimensions since $A$ is not row-equivalent to $B$.



      My approach for part (iii) seems a little too naive and I feel like there's more to the question than meets the eye.



      Help would be appreciated!










      share|cite|improve this question













      Given matrix $$A = begin{pmatrix} 1 & r & 1 & 1 \ 1 & s & 1 & 2 \ 1 & t & 1 & 3 \ 1&0&0&1end{pmatrix}$$ and let $B$ be the matrix obtained from $A$ by exchanging the $2^{nd}$ and $4^{th}$ columns, i.e. $$B = begin{pmatrix} 1&1&1&r\1&2&1&s\1&3&1&t\1&1&0&0 end{pmatrix}$$



      (i) Justify whether the column space of $A$ the same as the column space of $B$.



      (ii)Justify whether the row spaces of $A$ and $B$ have the same dimension.



      (iii) Let $C$ by an invertible 4-by-4 matrix. Determine a basis for the row space of the 4-by-8 matrix $$D = begin{bmatrix} mathbf{C} & mathbf{C}end{bmatrix}$$



      For part (i), this is my attempt:



      I note that $B^T$ is the matrix obtained from $A^T$ by swapping the $2^{nd}$ and $4^{th}$ rows of $A^T$, thus $B^T$ is row equivalent to $A^T$ since elementary row operations preserve the row space $-$ $A$ has the same column space as $B$.



      For part (ii), I base my attempt on my answer from (i). $A$ and $B$ have the same column space, so $A$ and $B$ must have the same dimension of column space. Since $$dimRan(A) = dimRan(A^T)$$ and $$dimRan(B) = dimRan(A)$$ we get that $dimRan(B^T) = dimRan(A^T)$ i.e. the row spaces of $A$ and $B$ indeed have the same dimension.



      For part (iii), I make use of the fact that $$rref(C) = begin{pmatrix} 1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1end{pmatrix}$$ so $$rref(D) = begin{bmatrix} 1&0&0&0&1&0&0&0\0&1&0&0&0&1&0&0\0&0&1&0&0&0&1&0\0&0&0&1&0&0&0&1end{bmatrix}$$



      Thus we can take a basis for the row space of $D$ to be $${begin{pmatrix}1\0\0\0\1\0\0\0end{pmatrix},begin{pmatrix}0\1\0\0\0\1\0\0end{pmatrix},begin{pmatrix}0\0\1\0\0\0\1\0end{pmatrix},begin{pmatrix}0\0\0\1\0\0\0\1end{pmatrix}}$$



      I'm pretty sure my answer for (i) is correct, but I'd just like to verify and see if there would be another way to explain it, but I'm unsure about part (ii) as intuitively, I would expect the row spaces of $A$ and $B$ to have different dimensions since $A$ is not row-equivalent to $B$.



      My approach for part (iii) seems a little too naive and I feel like there's more to the question than meets the eye.



      Help would be appreciated!







      linear-algebra matrices proof-verification proof-explanation






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      asked Nov 18 at 6:10









      uznam

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          Seems that you know what to do.



          For part $(i)$, the column space are spanned by the columns. Hence interchanging the columns doesn't change the space. Hence the column space is the same.



          For part $(ii)$, the dimension of the row space is equal to the dimension of the column space. Hence the dimension of the row space is equal.



          For part $(iii)$, if $P$ is invertible $PA$ and $A$ share the same row space. Hence for $D=[C, C]$. We can premultiply $D$ by $C^{-1}$ and obtain $[I, I]$. Hence a basis would be ${(e_i + e_{4+i})^T|i in { 1,2,3,4}}$ where $e_i in mathbb{R}^{8 times 1}$ is the $i$-th standard unit basis.






          share|cite|improve this answer





















          • Could you elaborate more on the intuition behind the fact that 'if $P$ is invertible then $PA$ and $A$ share the same row space' ? I have never actually seen this explicitly taught, so I'm a little lost.
            – uznam
            Nov 18 at 8:09






          • 1




            elementary row operations doesn't change the row space. Invertible matrix can be written as a product of elementary matrices.
            – Siong Thye Goh
            Nov 18 at 8:15











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          1 Answer
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          active

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          active

          oldest

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          active

          oldest

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          up vote
          1
          down vote



          accepted










          Seems that you know what to do.



          For part $(i)$, the column space are spanned by the columns. Hence interchanging the columns doesn't change the space. Hence the column space is the same.



          For part $(ii)$, the dimension of the row space is equal to the dimension of the column space. Hence the dimension of the row space is equal.



          For part $(iii)$, if $P$ is invertible $PA$ and $A$ share the same row space. Hence for $D=[C, C]$. We can premultiply $D$ by $C^{-1}$ and obtain $[I, I]$. Hence a basis would be ${(e_i + e_{4+i})^T|i in { 1,2,3,4}}$ where $e_i in mathbb{R}^{8 times 1}$ is the $i$-th standard unit basis.






          share|cite|improve this answer





















          • Could you elaborate more on the intuition behind the fact that 'if $P$ is invertible then $PA$ and $A$ share the same row space' ? I have never actually seen this explicitly taught, so I'm a little lost.
            – uznam
            Nov 18 at 8:09






          • 1




            elementary row operations doesn't change the row space. Invertible matrix can be written as a product of elementary matrices.
            – Siong Thye Goh
            Nov 18 at 8:15















          up vote
          1
          down vote



          accepted










          Seems that you know what to do.



          For part $(i)$, the column space are spanned by the columns. Hence interchanging the columns doesn't change the space. Hence the column space is the same.



          For part $(ii)$, the dimension of the row space is equal to the dimension of the column space. Hence the dimension of the row space is equal.



          For part $(iii)$, if $P$ is invertible $PA$ and $A$ share the same row space. Hence for $D=[C, C]$. We can premultiply $D$ by $C^{-1}$ and obtain $[I, I]$. Hence a basis would be ${(e_i + e_{4+i})^T|i in { 1,2,3,4}}$ where $e_i in mathbb{R}^{8 times 1}$ is the $i$-th standard unit basis.






          share|cite|improve this answer





















          • Could you elaborate more on the intuition behind the fact that 'if $P$ is invertible then $PA$ and $A$ share the same row space' ? I have never actually seen this explicitly taught, so I'm a little lost.
            – uznam
            Nov 18 at 8:09






          • 1




            elementary row operations doesn't change the row space. Invertible matrix can be written as a product of elementary matrices.
            – Siong Thye Goh
            Nov 18 at 8:15













          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          Seems that you know what to do.



          For part $(i)$, the column space are spanned by the columns. Hence interchanging the columns doesn't change the space. Hence the column space is the same.



          For part $(ii)$, the dimension of the row space is equal to the dimension of the column space. Hence the dimension of the row space is equal.



          For part $(iii)$, if $P$ is invertible $PA$ and $A$ share the same row space. Hence for $D=[C, C]$. We can premultiply $D$ by $C^{-1}$ and obtain $[I, I]$. Hence a basis would be ${(e_i + e_{4+i})^T|i in { 1,2,3,4}}$ where $e_i in mathbb{R}^{8 times 1}$ is the $i$-th standard unit basis.






          share|cite|improve this answer












          Seems that you know what to do.



          For part $(i)$, the column space are spanned by the columns. Hence interchanging the columns doesn't change the space. Hence the column space is the same.



          For part $(ii)$, the dimension of the row space is equal to the dimension of the column space. Hence the dimension of the row space is equal.



          For part $(iii)$, if $P$ is invertible $PA$ and $A$ share the same row space. Hence for $D=[C, C]$. We can premultiply $D$ by $C^{-1}$ and obtain $[I, I]$. Hence a basis would be ${(e_i + e_{4+i})^T|i in { 1,2,3,4}}$ where $e_i in mathbb{R}^{8 times 1}$ is the $i$-th standard unit basis.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 18 at 8:02









          Siong Thye Goh

          94.3k1462114




          94.3k1462114












          • Could you elaborate more on the intuition behind the fact that 'if $P$ is invertible then $PA$ and $A$ share the same row space' ? I have never actually seen this explicitly taught, so I'm a little lost.
            – uznam
            Nov 18 at 8:09






          • 1




            elementary row operations doesn't change the row space. Invertible matrix can be written as a product of elementary matrices.
            – Siong Thye Goh
            Nov 18 at 8:15


















          • Could you elaborate more on the intuition behind the fact that 'if $P$ is invertible then $PA$ and $A$ share the same row space' ? I have never actually seen this explicitly taught, so I'm a little lost.
            – uznam
            Nov 18 at 8:09






          • 1




            elementary row operations doesn't change the row space. Invertible matrix can be written as a product of elementary matrices.
            – Siong Thye Goh
            Nov 18 at 8:15
















          Could you elaborate more on the intuition behind the fact that 'if $P$ is invertible then $PA$ and $A$ share the same row space' ? I have never actually seen this explicitly taught, so I'm a little lost.
          – uznam
          Nov 18 at 8:09




          Could you elaborate more on the intuition behind the fact that 'if $P$ is invertible then $PA$ and $A$ share the same row space' ? I have never actually seen this explicitly taught, so I'm a little lost.
          – uznam
          Nov 18 at 8:09




          1




          1




          elementary row operations doesn't change the row space. Invertible matrix can be written as a product of elementary matrices.
          – Siong Thye Goh
          Nov 18 at 8:15




          elementary row operations doesn't change the row space. Invertible matrix can be written as a product of elementary matrices.
          – Siong Thye Goh
          Nov 18 at 8:15


















           

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