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This document contains math tutorial questions and solutions for applying theorems about perpendicular bisectors, angle bisectors, the Pythagorean theorem, and classifying triangles by their side lengths. The questions cover finding side lengths, angle measures, and determining if sets of values could represent triangle sides. The solutions show the step-by-step work using the appropriate theorems.
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-1 3 3 1
3)
Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.
4) u =
5
-3
5
, A =
1 -3 4
-1 0 -5
3 -3 6
A) In Col A and in Nul A B) In Col A, not in Nul A
C) Not in Col A, in Nul A D) Not in Col A, not in Nul A
4)
Use coordinate vectors to determine whether the given polynomials are linearly dependent in P2. Let B be the standard
basis of the space P2 of polynomials, that is, let B = 1, t, t2 .
5) 1 + 2t, 3 + 6t2, 1 + 3t + 4t2
A) Linearly dependent B) Linearly independent
5)
Find the dimensions of the null space and the column space of the given matrix.
6) A = 1 -5 -4 3 0
-2 3 -1 -4 1
A) dim Nul A = 2, dim Col A = 3 B) dim Nul A = 4, dim Col A = 1
C) dim Nul A = 3, dim Col A = 2 D) dim Nul A = 3, dim Col A = 3
6)
1
Solve the problem.
7) Let H =
a + 3b + 4d
c + d
-3a - 9b + 4c - 8d
-c - d
: a, b, c, d in ℛ
Find the dimension of the subspace H.
A) dim H = 3 B) dim H = 1 C) dim H = 4 D) dim H = 2
7)
Assume that the matrix A is row equivalent to B. Find a basis for the row space of the matrix A.
8) A =
1 3 -4 0 1
2 4 -5 5 -2
1 -5 0 -3 2
-3 -1 8 3 -4
, B =
1 3 -4 0 1
0 -2 3 5 -4
0 0 -8 -23 17
0 0 0 0 0
A) {(1, 3, -4, 0, 1), (0, -2, 3, 5, -4), (0, 0, -8, -23, 17), (0, 0, 0, 0, 0)}
B) {(1, 3, -4, 0, 1), (0, -2, 3, 5, -4), (0, 0, -8, -23, 17)}
C) {(1, 3, -4, 0, 1), (2, 4, -5, 5), -2, (1, ...
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