20f-sample-mt2

20f-sample-mt2 - bases of R 2 : B = { e 1 + e 2 , 2 e 2 }...

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MATH. 20F SAMPLE MIDTERM 2 You have 50 minutes for this exam. Please write legibly and show all working. No calculators are allowed. Write your name and ID number. Name: ID Number: (1i) (5 points) Find the determinant of A = 1 0 0 1 2 0 1 0 11 3 7 5 0 0 - 1 0 . (ii) (5 points) If A is an n × n matrix, how are det(3 A ) and det( A ) related? 1
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2 MATH. 20F SAMPLE MIDTERM 2 (2i) (10 points) Let P 2 be the vector space of polynomials of degree 2, and let W = { p P 2 : p (0) + p (1) = 0 } . Show that W is a subspace of P 2 and Fnd a basis for W . (ii) (10 points) Consider the map T : P 2 P 2 deFned by T ( p ) = p ′′ ( t ) + p (0) . Show that T is linear and determine its range.
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MATH. 20F SAMPLE MIDTERM 2 3 (3) Let { e 1 ,e 2 } be the standard basis of R 2 . Consider the following two
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Unformatted text preview: bases of R 2 : B = { e 1 + e 2 , 2 e 2 } and C = { e 1-e 2 ,e 1 } . (i) (5 points) Find the change of cooridnate matrix from B-coordinates to C-coordinates. (ii) (5 points) If a vector w is such that [ w ] B = (1 , 3) T , what is [ w ] C ? 4 MATH. 20F SAMPLE MIDTERM 2 (4) (10 points) Decide if the following statements are true or false. Justify your answers. (a) If A is a 6 7 matrix whose row space has dimension 2, then the null space of A has dimension 5. (b) If A and B are related by a sequence of EROs, then the column space of A and B are equal....
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20f-sample-mt2 - bases of R 2 : B = { e 1 + e 2 , 2 e 2 }...

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