MTH 1230 -- Linear Algebra -- Fall 1997

Quiz 2

Problem 1.
- (a) Find the LU-decomposition of $A=\bmatrix 3 & 0 & 1 \\ -6 & 4 & -3 \\ 3 & 2 & 1 \endbmatrix.$
- (b) Use the LU-decomposition of A from part (a) to solve $A\cdot \bmatrix x_1\\ x_2\\ x_3 \endbmatrix = \bmatrix 3\\ 4\\ 2 \endbmatrix.$
Problem 2. Use Gaussian elimination to find the inverse of the matrix $A=\bmatrix 1&0&1\\ 2&2&0\\ 1&1&1\endbmatrix.$
Problem 3. Consider the matrix
- (a) Find elementary matrices E₁, E₂, and E₃ such that E₃E₂E₁A = I.
- (b) Use part (a) to express A as a product of elementary matrices.
- (c) Use part (a) to express A^-1 as a product of elementary matrices.
- (d) Use part (c) to find A^-1.
Problem 4. Let
- (a) Find unit vectors in the direction of a and b, respectively.
- (b) Find the lengths of a and b, and compute the dot product a.b.
- (c) Find the angle between a and b. Are a and b orthogonal?
- (d) Find the projection of a onto b.
- (e) Find the projection of b and a.

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Created by Alexandru Suciu, Fri Oct 17, 1997

http://www.math.neu.edu/~suciu/mth1230/1230f97.q2/index.html