Assign1-2004

# Assign1-2004 - A v 1,A v n is a basis for R n b Let T be a...

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Linear Algebra II. MATH 2107 Assignment No1 Due date: February 11 1) Suppose u = x y z and v = z 0 y . a. Find a 3 × 3 matrix A such that A u = v . b. Is matrix A invertible? Justify your answer. 2) suppose A = 1 3 0 2 5 1 - 3 - 9 - 1 and B = 4 3 3 x - 1 - 1 y z - 1 such that A - 1 = B a. Find x,y and z . b. Find inverse of C = 1 2 - 3 3 5 - 9 0 1 - 1 . 1

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3) Let B = 0 - 1 0 - 1 0 5 0 2 - 1 2 - 2 - 1 0 0 0 1 a. Find all values for t such that the determinant of ( B - tI 4 ) is zero. Justify your answer. [Hint: use some cofactors! Use them repeatedly. Note, for example, the fourth row is almost all zero.] b. Find all values for t such that ( B - tI 4 ) T is not invertible. Justify your answer. 4) Using row operations, show that if A = 1 x 2 x 4 1 y 2 y 4 1 z 2 z 4 , then det A = ( y 2 - x 2 )( z 2 - x 2 )( z 2 - y 2 ) . 5) Find the dimensions of the following subspaces of R 4 . a. All vectors of the form ( a,b,c,d ) with d = a - b . b. All vectors of the form ( a,b,c,d ) with c = a + b and d = a - b . c. All vectors of the form ( a,b,c,d ) with a = d . d. All vectors of the form ( a - c,a - b, - b + c, - a - b ). 2
6) a. Suppose that { v 1 ,..., v n } is a basis for R n . Show that if A is an n × n invertible matrix, than
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Unformatted text preview: { A v 1 ,...,A v n } is a basis for R n . b. Let T be a linear transformation on R n and let e 1 , e 2 ,..., e n be the standard basis. Let A be the standard matrix of T . Prove that if T ( e 1 ) ,T ( e 2 ) ,...,T ( e n ) is a basis of R n then A is invertible. 7) Let B = 1 2 3 4-1-2-4-6 4 8 a. Find a basis for the row space of A . b. Find a basis for the column space of A . c. Find a basis for the null space of A . d. What is the rank of A ? 8) a. Let H and K be subspaces of vector space V . The intersection of H and K , written as H ∩ K , is the set of v in V that belong to both H and K . Show that H ∩ K is a subspace of V . b. Give an example to show that the union of two subspaces is not, in general, a subspace. Justify your answer. 3...
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Assign1-2004 - A v 1,A v n is a basis for R n b Let T be a...

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