Linear Algebra Answers

Suppose A and B are n x n matrices with A invertible. Prove that det ABA^-1 = det B

Let u = (4,2,-1), v = (3, 1, 1) and w = (0, 2, 1). Compute the following:

(i) 2v - 3w -u

(ii) u(w+v)

(iii) ||u . w ||

(iv) the orthogonal projection of u on w

(v) the vector component of u orthogonal to w

1. Verify if the vectors (3, 4, 5), (-3, 0, 5), (4, 4, 4), (3, 4, 0) are linearly independent.

2. Let A={b1,b2,b3} be a set of three-dimensional vectors in R3.

a. Prove that if the set A is linearly independent, then A is a basis of the vector space R3.

b. Prove that if the set A spans R3, then A is a basis of R3.

a students is taking five course and is facing the crunch of final exams.she estimates that she has 40 hours available to study .if xj=the number of hours allocated to study for coursej.state the equation whose solution set specifies all possible allocation of time among the five courses which will exhaust the 40 hours

how that the set of all 3 x 3 upper

triangular matrices with entries from R is a

vector space over R under usual addition

and scalar multiplication of matrices. Find a

basis of this vector space.

Evaluate det (A) by expanding along the 2nd row.

Determine the area of the parallelogram determined by u(1,0) and v(0,1)

Solve the following system of linear equations by using the inverse matrix

method: (10)

x + y + z = 4

-2x - y + 3z = 1

y + 5z = 9

Find

a)-A raised to -1+3B raised to T

b)B raised to -1+(A raised to T+A raised to -1)

Let a11 x1 + a12 x2 + a13x3 = b1

a21 x1 + a22 x2 + a23 x3 = b2

a31 x1 + a32 x2 + a33 x3 = b3.

Show that if det (A) 6= 0 where det(A) is the determinant of the coefficient matrix;

then x2 = det(A2)

det(A) where det(A2) is the determinant obtained by replacing the second column of det(A)

by (b1; b2; b3)T :

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