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These are the notes of Exam of Linear Algebra which includes Initial Value Problem, General Solution, Erential Equation, Origin Parallel, Line, Vector Space, Dimension etc. Key important points are: Linear, Matrix, Transformation, Standard Matrix, One to One, Vector, Matrix Operations, Standard Matrix, Inverse, Invertible
Typology: Exams
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Define the linear transformation T : R^3 Ï R^3 so that
x 1 x 2 x 3
x 1 − x 2 + 2 x 3 −x 1 − 2 x 2 + x 3 2 x 1 + x 2 + x 3
a.) Find the standard matrix of T. b.) Is T one-to-one? Explain. c.) Is T onto? Explain.
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^3 so that
x 1 x 2 x 3 x 4
[ (^) x 1 −^ x 2 +^ x 3 +^ x 4 x 1 + x 2 + x 3 − x 4
and S
([ (^) x 1 x 2
−x 1 + x 2 x 1 − 2 x 2 2 x 1 − x 2
a.) Find the standard matrix of S ◦ T.
b.) Find, if there is any, a vector v⃗ such that ( S ◦ T )( v⃗ ) = b⃗ where ⃗b =
Use the invertible matrix theorem to determine the value(s) of λ for which the matrix
A =
1 λ 0 3 2 0 1 2 1
is NOT invertible.
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 2 − x 3 x 1 + x 4
a.) Find the column space of T. Find the dimension of the column space. b.) Find the null space of T. Find the dimension of the null space. c.) Find a basis for the column space of T. d.) Find a basis for the null space of T .(Remark. The column space of T is Col A and the null space of T is Nul A where A is the standard matrix of T .)
