{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

laex2 - MAS 4105 Test 2 1(8 pts In the following let V W...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAS 4105 Test 2 1. (8 pts) In the following let V, W, and U be a vector spaces of finite dimension over a field F with ordered bases β, γ, and δ respectively. Indicate whether the following are true or false. i. If T : V → W is linear and dim( V ) = dim( W ), then T is invertible if and only if [ T ] γ β is invertible. T F ii. Let T : V → W be a linear transformation and x ∈ V ; then [ T ( x )] γ = [ T ] γ β [ x ] β . T F iii. If T : V → W is linear, then T is one-to-one if and only if T is onto. T F iv. If T : V → W and S : W → U are linear, then [ ST ] δ β = [ S ] δ β [ T ] γ β . T F v. If T : V → W and S : W → U are linear and if x ∈ Nul( T ), then x ∈ Nul( ST ). T F vi. Let T : V → W be a transformation, then T is linear if and only if T ( ax + y ) = aT ( x )+ T ( y ) for all x, y ∈ V and a ∈ F . T F vii. Let T : W → U be a linear transformation, then R ( T ) = span( T ( γ )). T F viii. Let S : U → W be an invertible linear transformation, then nullity( S- 1 ) + rank( S- 1 ) = dim( U ). T F 2. (10 pts) Let T : R 3 → R 3 be defined by T a b c = a + b + c a + b c . i. Prove that T is linear. ii. What is the Null space for T and what is its dimension? iii. Use the Dimension Theorem to determine the rank of T . 3. (15 pts) Let β = { 1 , x, x 2 } be an ordered basis for P 2 ( R ) and let γ = 1 , 1 1 , 1 1 1 be an ordered basis for R 3 . Define the invertible, linear transformation T : P 2 ( R ) → R 3 by T ( ax 2 + bx + c ) = a + c a + b a ....
View Full Document

{[ snackBarMessage ]}

Page1 / 10

laex2 - MAS 4105 Test 2 1(8 pts In the following let V W...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online