MATH 0280 MIDTERM I REVIEW, Fall 2023; Cover §1.1–§3.5
1. Vectors and Geometry
Let A = (0, 1, 3), B = (0, 6, 6), C = (3, −5, 5), and O = (0, 0, 0).
−→
−−→
(a) Let u = AB, v = BC. (1) Simplify 2u + 3v; (2) Find the angel 6 ABC; (3) Find
the area of 4ABC.
(b) Find a vector equation for the line that passes through A and B; also find the
distance from C to the line.
(c) Find a linear equation for the plane that passes through A,B, and C; also find a
−−→
point D on the plane such that OD is perpendicular to the plane.
2. Linear Equations
(a) Let k be a constant. For unknown x, y, z, solve the system x+y+z = 1, 2x+y+z =
5, 6x + y + z = k.
(b) Let k be a constant. For unknown x, y, z, solve the system x + y + kz = 1, x +
ky + z = 1, kx + y + z = −2.
(c) Show that u, v, w is in span{u + v, 2u + 3v, 4v + 6w}.
(d) Check the linear dependency of the matrices B,C,D in the next problem.
(e) Let v1 = [1 2 3], v2 = [2 5 6], u = [2 1 0] and w = [1 1 1]. Is u ∈ span(v1 , v2 )?
Is w ∈ span(v1 , v2 )? If yes, write u or w as a linear combination of v1 and v2 .
3. Matrix Algebra
(a) Let
"
A=
1 2 3
4 5 6
#
"
,B=
1 2
4 5
#
"
,C=
6 7
8 9
#
"
,D=
1 3
0 0
#

Find (1) AAT and AT A; (2) Find B −1 and I − 6B − 3B 2 ; (2) Solve matrix X
from XB − CX = D. (3) Find E −1 .
1
1 3 1
6
 2 −1 0 1 −1 


(b) Let A = 
 . Find bases for row(A), col(A), and null(A);
 −3 2 1 −2 1 
4
1 6 1
3
also find rank(A) and nullity(A).


1

1 2 −1


,E =  2 2 4 .
1 3 −3