Rank and Nullity of linear transformationDetermining Rank and NullityThe computation of nullity and rank of a linear transformation.Relation between a matrix and transformation.Proving that the rank and nullity of a transformation equals the rank and nullity of a left multiplication transformationNullity of linear transformationrank and nullity of a linear transformationNullity and rank of a linear transformation over a function spaceNullity and Rank of a Linear TransformationFind rank and nullity of this linear transformation.How to see the Image, rank, null space and nullity of a linear transformation
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Rank and Nullity of linear transformation
Determining Rank and NullityThe computation of nullity and rank of a linear transformation.Relation between a matrix and transformation.Proving that the rank and nullity of a transformation equals the rank and nullity of a left multiplication transformationNullity of linear transformationrank and nullity of a linear transformationNullity and rank of a linear transformation over a function spaceNullity and Rank of a Linear TransformationFind rank and nullity of this linear transformation.How to see the Image, rank, null space and nullity of a linear transformation
$begingroup$
Let B be any non-zero fixed 2×2 matrix and T be a transformation from M 2×2 to M 2×2 defined by T(A)=BA for A€M 2×2 .
If det(B)=0 then find the rank and Nullity of T.
I am getting answer as rank=1, Nullity =1 by taking B for rank and Nullity. But how to find rank and Nullity of T.
linear-algebra linear-transformations
asked Mar 14 at 8:24
user90596user90596
596
$endgroup$
add a comment |
$begingroup$
Let B be any non-zero fixed 2×2 matrix and T be a transformation from M 2×2 to M 2×2 defined by T(A)=BA for A€M 2×2 .
If det(B)=0 then find the rank and Nullity of T.
I am getting answer as rank=1, Nullity =1 by taking B for rank and Nullity. But how to find rank and Nullity of T.
linear-algebra linear-transformations
asked Mar 14 at 8:24
user90596user90596
596
$endgroup$
$begingroup$
If $B=O$ then $text rank (T) = 0, text Nullity (T) = 4.$
$endgroup$
– Dbchatto67
Mar 14 at 8:31
$begingroup$
det(B)=0 not B=0
$endgroup$
– user90596
Mar 14 at 8:36
$begingroup$
If $B neq O$ then $text rank (B) = 1.$ So $text Nullity (B) = 1$ by rank nulity theorem. But then $text rank (T) = text Nullity (T) = 2.$
$endgroup$
– Dbchatto67
Mar 14 at 8:37
add a comment |
$begingroup$
Let B be any non-zero fixed 2×2 matrix and T be a transformation from M 2×2 to M 2×2 defined by T(A)=BA for A€M 2×2 .
If det(B)=0 then find the rank and Nullity of T.
I am getting answer as rank=1, Nullity =1 by taking B for rank and Nullity. But how to find rank and Nullity of T.
linear-algebra linear-transformations
asked Mar 14 at 8:24
user90596user90596
596
$endgroup$
Let B be any non-zero fixed 2×2 matrix and T be a transformation from M 2×2 to M 2×2 defined by T(A)=BA for A€M 2×2 .
If det(B)=0 then find the rank and Nullity of T.
I am getting answer as rank=1, Nullity =1 by taking B for rank and Nullity. But how to find rank and Nullity of T.
linear-algebra linear-transformations
linear-algebra linear-transformations
asked Mar 14 at 8:24
user90596user90596
596
asked Mar 14 at 8:24
user90596user90596
596
asked Mar 14 at 8:24
user90596user90596
596
asked Mar 14 at 8:24
user90596user90596
596
asked Mar 14 at 8:24
user90596user90596
596
596
$begingroup$
If $B=O$ then $text rank (T) = 0, text Nullity (T) = 4.$
$endgroup$
– Dbchatto67
Mar 14 at 8:31
$begingroup$
det(B)=0 not B=0
$endgroup$
– user90596
Mar 14 at 8:36
$begingroup$
If $B neq O$ then $text rank (B) = 1.$ So $text Nullity (B) = 1$ by rank nulity theorem. But then $text rank (T) = text Nullity (T) = 2.$
$endgroup$
– Dbchatto67
Mar 14 at 8:37
add a comment |
$begingroup$
If $B=O$ then $text rank (T) = 0, text Nullity (T) = 4.$
$endgroup$
– Dbchatto67
Mar 14 at 8:31
$begingroup$
det(B)=0 not B=0
$endgroup$
– user90596
Mar 14 at 8:36
$begingroup$
If $B neq O$ then $text rank (B) = 1.$ So $text Nullity (B) = 1$ by rank nulity theorem. But then $text rank (T) = text Nullity (T) = 2.$
$endgroup$
– Dbchatto67
Mar 14 at 8:37
$begingroup$
If $B=O$ then $text rank (T) = 0, text Nullity (T) = 4.$
$endgroup$
– Dbchatto67
Mar 14 at 8:31
$begingroup$
If $B=O$ then $text rank (T) = 0, text Nullity (T) = 4.$
$endgroup$
– Dbchatto67
Mar 14 at 8:31
$begingroup$
det(B)=0 not B=0
$endgroup$
– user90596
Mar 14 at 8:36
$begingroup$
det(B)=0 not B=0
$endgroup$
– user90596
Mar 14 at 8:36
$begingroup$
If $B neq O$ then $text rank (B) = 1.$ So $text Nullity (B) = 1$ by rank nulity theorem. But then $text rank (T) = text Nullity (T) = 2.$
$endgroup$
– Dbchatto67
Mar 14 at 8:37
$begingroup$
If $B neq O$ then $text rank (B) = 1.$ So $text Nullity (B) = 1$ by rank nulity theorem. But then $text rank (T) = text Nullity (T) = 2.$
$endgroup$
– Dbchatto67
Mar 14 at 8:37
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Since $Bneq0$ and $det(B)=0$, there are linearly independent vectors $v_1=(a_11,a_12)$ and $v_2=(a_21,a_22)$ such that $B.v_1=(a,b)neq(0,0)$ and that $B.v_2=(0,0)$. Consider the matrices$$E_1=beginbmatrixa_11&a_11\a_12&a_12endbmatrix, E_2=beginbmatrixa_11&a_21\a_12&a_22endbmatrix, E_3=beginbmatrixa_21&a_22\a_21&a_22endbmatrixtext, and E_4=beginbmatrixa_21&a_11\a_22&a_12endbmatrix.$$Then $E_1,E_2,E_3,E_4$ is a basis of $M_2times2$. Compute $T(E_i)$ for each $iin1,2,3,4$. We will see that $dim T(M_2times2)=2$. Then apply the rank-nullity theorem.
answered Mar 14 at 8:39
José Carlos SantosJosé Carlos Santos
169k23132237
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Since $Bneq0$ and $det(B)=0$, there are linearly independent vectors $v_1=(a_11,a_12)$ and $v_2=(a_21,a_22)$ such that $B.v_1=(a,b)neq(0,0)$ and that $B.v_2=(0,0)$. Consider the matrices$$E_1=beginbmatrixa_11&a_11\a_12&a_12endbmatrix, E_2=beginbmatrixa_11&a_21\a_12&a_22endbmatrix, E_3=beginbmatrixa_21&a_22\a_21&a_22endbmatrixtext, and E_4=beginbmatrixa_21&a_11\a_22&a_12endbmatrix.$$Then $E_1,E_2,E_3,E_4$ is a basis of $M_2times2$. Compute $T(E_i)$ for each $iin1,2,3,4$. We will see that $dim T(M_2times2)=2$. Then apply the rank-nullity theorem.
answered Mar 14 at 8:39
José Carlos SantosJosé Carlos Santos
169k23132237
$endgroup$
add a comment |
$begingroup$
Since $Bneq0$ and $det(B)=0$, there are linearly independent vectors $v_1=(a_11,a_12)$ and $v_2=(a_21,a_22)$ such that $B.v_1=(a,b)neq(0,0)$ and that $B.v_2=(0,0)$. Consider the matrices$$E_1=beginbmatrixa_11&a_11\a_12&a_12endbmatrix, E_2=beginbmatrixa_11&a_21\a_12&a_22endbmatrix, E_3=beginbmatrixa_21&a_22\a_21&a_22endbmatrixtext, and E_4=beginbmatrixa_21&a_11\a_22&a_12endbmatrix.$$Then $E_1,E_2,E_3,E_4$ is a basis of $M_2times2$. Compute $T(E_i)$ for each $iin1,2,3,4$. We will see that $dim T(M_2times2)=2$. Then apply the rank-nullity theorem.
answered Mar 14 at 8:39
José Carlos SantosJosé Carlos Santos
169k23132237
$endgroup$
add a comment |
$begingroup$
Since $Bneq0$ and $det(B)=0$, there are linearly independent vectors $v_1=(a_11,a_12)$ and $v_2=(a_21,a_22)$ such that $B.v_1=(a,b)neq(0,0)$ and that $B.v_2=(0,0)$. Consider the matrices$$E_1=beginbmatrixa_11&a_11\a_12&a_12endbmatrix, E_2=beginbmatrixa_11&a_21\a_12&a_22endbmatrix, E_3=beginbmatrixa_21&a_22\a_21&a_22endbmatrixtext, and E_4=beginbmatrixa_21&a_11\a_22&a_12endbmatrix.$$Then $E_1,E_2,E_3,E_4$ is a basis of $M_2times2$. Compute $T(E_i)$ for each $iin1,2,3,4$. We will see that $dim T(M_2times2)=2$. Then apply the rank-nullity theorem.
answered Mar 14 at 8:39
José Carlos SantosJosé Carlos Santos
169k23132237
$endgroup$
Since $Bneq0$ and $det(B)=0$, there are linearly independent vectors $v_1=(a_11,a_12)$ and $v_2=(a_21,a_22)$ such that $B.v_1=(a,b)neq(0,0)$ and that $B.v_2=(0,0)$. Consider the matrices$$E_1=beginbmatrixa_11&a_11\a_12&a_12endbmatrix, E_2=beginbmatrixa_11&a_21\a_12&a_22endbmatrix, E_3=beginbmatrixa_21&a_22\a_21&a_22endbmatrixtext, and E_4=beginbmatrixa_21&a_11\a_22&a_12endbmatrix.$$Then $E_1,E_2,E_3,E_4$ is a basis of $M_2times2$. Compute $T(E_i)$ for each $iin1,2,3,4$. We will see that $dim T(M_2times2)=2$. Then apply the rank-nullity theorem.
answered Mar 14 at 8:39
José Carlos SantosJosé Carlos Santos
169k23132237
answered Mar 14 at 8:39
José Carlos SantosJosé Carlos Santos
169k23132237
answered Mar 14 at 8:39
José Carlos SantosJosé Carlos Santos
169k23132237
answered Mar 14 at 8:39
José Carlos SantosJosé Carlos Santos
169k23132237
169k23132237
add a comment |
add a comment |
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$begingroup$
If $B=O$ then $text rank (T) = 0, text Nullity (T) = 4.$
$endgroup$
– Dbchatto67
Mar 14 at 8:31
$begingroup$
det(B)=0 not B=0
$endgroup$
– user90596
Mar 14 at 8:36
$begingroup$
If $B neq O$ then $text rank (B) = 1.$ So $text Nullity (B) = 1$ by rank nulity theorem. But then $text rank (T) = text Nullity (T) = 2.$
$endgroup$
– Dbchatto67
Mar 14 at 8:37