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Thes are the notes of Exam of Linear Algebra which includes Sufficient, Given System, Solution Exists, Suppose, Matrix, Eigenvalues, Corresponding Eigenvectors etc. Key important points are: Vector, Standard Matrix, Matrix, Linear Transformation, One to One, Matrix Operations, Mention, Inverse of a Matrix, Invertible Linear Transformation, Characterizations
Typology: Exams
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Define the linear transformation T : R
4 Ï R
3 so that
x 1
x 2
x 3
x 4
x 1 + x 2 − x 3 + x 4
2 x 1 + x 2 + 4 x 3 + x 4
3 x 1 + x 2 + 9 x 3
a.) Find the standard matrix of T.
b.) Is T one-to-one?
c.) Is T onto?
d.) If there is any, find a vector v ⃗ such that Tv ( ⃗ ) = ⃗b where b⃗ =
Define the linear transformations T : R 4 Ï R 2 and S : R 2 Ï R 4 so that
x 1
x 2
x 3
x 4
x 1 − x 2 + x 3 + x 4
−x 1 + x 2 + 2 x 3 − 2 x 4
and S
x 1
x 2
−x 1
−x 1 + 2 x 2
2 x 1 − 2 x 2
x 2
a.) Find the standard matrix of S ◦ T.
b.) Is it possible to mention T ◦ S? Explain.
c.) Find, if there is any, a vector v⃗ such that ( S ◦ T )( v⃗ ) = ⃗b where b⃗ =
a.) Use the invertible matrix theorem to determine the value(s) of λ for which the matrix
1 λ 0
1 1 1
0 0 1
is invertible.
b.) For those values found in part (a.) find the inverse of A.
n
Define the linear transformation T : R 4 Ï R 3 by
T ( x 1 , x 2 , x 3 , x 4 ) = ( x 1 + x 2 − x 3 + x 4 , 2 x 1 + x 2 + 4 x 3 + x 4 , 3 x 1 + x 2 + 9 x 3 ).
Find a basis for the null space of T .(Remark. The null space of T is the null space of A where A is the
standard matrix of T .)
