Matrix Algebra
Existence & Uniqueness
Basics
Quick Problem Solving
Transformations
100
A matrix that is row equivalent to the n x n identity matrix & A has n pivot positions
What is an invertible matrix?
100
(i) a unique solution (no free variables) or
(ii) infinitely many solutions
What is a consistent solution?
100
Replacement, Interchange, Scaling
What are the three elementary row operations?
100
For what values of h & k is the system consistent?
2x1-x2=h
-6x1+3x2=k
When k + 3h = 0
100
(i) T(u+v)=T(u)+T(v) for all u,v in the domain of T
(ii)T(cu)=cT(u) for all scalars c and in the domain of T
What is a linear transformation?
200
It is equal to B-1A-1
What is the (AB)-1?
200
It is a set of outputs in Rm
What is a Co-domain?
200
A type of matrix where the leading entry in each nonzero row is 1 & each leading 1 is the only nonzero entry in its column.
What is an echelon matrix?
200
Construct a 3x3 matrix A and vectors b and c in R3
so that Ax = b has a solution, but Ax = c does not.
Many solutions are possible.
200
Transformation = [01 0-1]
What is a reflection through the x1 axis?
300
The dimension of the column space of A does not span n & the columns of A do not span Rn.
What is a non-invertable matrix?
300
Every b in the co-domain can be written as a linear combination of the columns of A. (E/U)
What is a marker of solution existence?
300
A solution when both Ax = 0 and there isn't a free variable.
What is a trivial solution?
300
Is the following matrix linearly independent?
| -4 -3 0 |
| 0 -1 4 |
| 1 0 3 |
| 5 4 6 |
Yes it is
300
A transformation with the standard matrix [k010]
What is a horizontal contraction and expansion?
400
(T/F) If A is invertible, then elementary row operations that reduce A to the identity In, also reduce A-1 to In
What is false?
400
T is one-to-one if and only if the columns of A are linearly dependent. (T/F)
False
400
A type of square n x n matrix whose nondiagonal entries are zero.
What is a diagonal matrix?
400
Let T: R2 -> R2 be the transformation that first performs a horizontal shear that maps e2 into e2 - 0.5e1 (but leaves e1 unchanged) and then reflects the result through the x2-axis. Assuming that T is linear, find its standard matrix. [Hint: Determine the final location of the images of e1 and e2.]
[-10.501] (see 1.9)
400
Each b in Rm is the image of at most one in Rn.
What is a one-to-one transformation?
500
Name 3 statements in the IMT:
Examples:
1. 
is row-equivalent to the 
identity matrix
2. 
has 
pivot positions.
3. The equation 
has only the trivial solution 
.
500
If each B in Rm is the image of at least one in Rn
What does it mean for a mapping to be onto?
500
A name for a matrix with no inverse.
What is a singular matrix?
500
If A is an invertible matrix, prove that 5A is an invertible matrix.
Since A is an invertible matrix, there exists a matrix C such that AC = I = CA. The goal is to find a matrix D so that (5A)D = I = D(5A). Set D = 1/5 C. Applying Theorem 2 from Section 2.1 establishes that (5A)(1/5 C) = (5)(1/5)(AC) = 1 I = I , and (1/5C)(5A) = (1/5)(5)(CA) = 1 I = I . Thus 1/5 C is indeed the inverse ofA, proving that A is invertible.
500
A transformation with the standard matrix [1k01]
What is a vertical shear?