For the given matrix B=
1
1
1
3
2
-2
4
3
-1
6
5
1
a.) Find a basis for the row space of matrix B.
b.) Find a basis for the column space of matrix B.
c.)Find a basis for the null space of matrix B.
d.) Find the rank and nullity of the matrix B.
Find a matrix representation of transformation T(x)=
2x1w1+x2w2-3x3w3 from R3 to a vector space W, where w1,w2, and w3 ∈
W. Clearly state how this matrix is representing the
transformation.
Let x ∈ R3 be nonzero and let
A be the matrix whose columns are x, 2x,
3x in this order. Show that x is an eigenvector of A and
find a basis for the null space of A.
T::R2->R2, T(x1,x2) =(x-2y,2y-x). a) verify that this
function is linear transformation. b)find the standard matrix for
this linear transformation. Determine the ker(T) and the range(T).
D) is this linear combo one to one? how about onto? what else could
we possibly call it?
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Find the transition matrix from the basis B =
{(2,1,0),(1,0,0)(0,1,1)} to the basis B'
={1,1,2),(1,1,1),(0,1,2)}
T(1+2x)=1+x-x^2
T(1-x^2)=2-x
T(1-2x+x^2)=3x-2x^2
a)compute T(-6x+3x^2)
b) find basis for N(T), null space of T
c) compute rank of T and find basis of R(T)
For the following matrices, first find a basis for the column
space of the matrix. Then use the Gram-Schmidt process to find an
orthogonal basis for the column space. Finally, scale the vectors
of the orthogonal basis to find an orthonormal basis for the column
space.
(a) [1 1 1, 1 0 2, 3 1 0, 0 0 4 ] b) [?1 6 6, 3 ?8 3, 1 ?2 6, 1
?4 ?3 ]