MAT 1341: LINEAR EQUATIONS I
An equation of the form
a1 x1 + a2 x2 + + an xn = b
(1)
is called a linear equation, where x1 , x2 , , xn are variables (unknowns) and b, a1 , a2 , , an
R. Here a1 , a2 , , an are called the coecients of the variables x1 , x2
MAT 1341: MATRICES II
1. Rank
Let A be a matrix. Recall that the number of leading 1s in a REF of A is equal to the
number of leading 1s in the RREF of A when we perform Gaussian elimination.
Denition 1.1. Let A be a matrix. The number of leading 1s in a
MAT 1341: DETERMINANTS I
Given an n n matrix A = [aij ], we denote by Aij the (n 1) (n 1) matrix obtained by
deleting ith row and jth column of A.
a11 a1 j1
a1j
a1 j+1 a1n
.
.
.
.
.
.
.
.
.
.
.
.
ai1 1 ai1 j1 ai1 j ai1 j+1 ai1 n
ain
aij
ai j+1
ai
MAT 1341: EIGENVECTORS AND EIGENVALUES I
[
]
a 0
Suppose that we have a 22 diagonal matrix D =
for some scalars a, b R. Then one
0 b
]
[ k
a 0
for any k 1. Similarly, for any n n diagonal matrix
can easily show that Dk =
0 bk
D whose diagonal entries are
MAT 1341: EIGENVECTORS AND EIGENVALUES II
1. Diagonalization
Recall that an n n matrix A is diagonalizable if[and only if P 1 AP = D for an invertible
]
matrix P , where D is a diagonal matrix. Let P = X1 X2 Xn , where X1 ,X2 , ,Xn
are columns of P . The
MAT 1341: INNER PRODUCT SPACE II
1. Orthogonal complement
Denition 1.1. Let W be a subspace of Rn . The orthogonal complement of W , denoted by
W , is the set of all vectors that are orthogonal to every vector in W :
W = cfw_v Rn | v w = 0 for all w W
Fo
MAT 1341: INNER PRODUCT SPACE I
A (real) inner product space is a subspace of Rn which has an inner product(=dot product).
1. Orthogonality
Denition 1.1. Let B = cfw_v1 , v2 , , vk be a set of nonzero vectors in Rn .
(1) B is called an orthogonal set if
MAT 1341: MATRICES I
1. Matrix multiplication
As we see in lecture note: Vector spaces I (page 2-3), the set of all m n matrices Mm,n (R)
forms a vector space. Here we introduce another operation which is called matrix multiplication
(or product of matric
MAT 1341: MATRICES III
1. Matrix Inverses
Denition 1.1. An n n matrix A (a square matrix) is called invertible if there is an n n
matrix B such that
AB = BA = In .
In this case, B is called the inverse of A, and is denoted by A1 , i.e., B = A1 .
Example 1
MAT 1341: LINEAR TRANSFORMATIONS
For any m n matrix A, consider a function (or map) T : Rn Rm given by T (x) = Ax.
Then, for any vectors x, y Rn and any scalar k R, we have
T (x + y) = A(x + y) = Ax + Ay = T (x) + T (y)
and
T (kx) = A(kx) = k(Ax) = k(T x)
MAT 1341: LINEAR EQUATIONS II
1. Gaussian Elimination
As we see in Example 0.7 of MAT 1341: Linear equations I, we can use matrices to solve a
system of linear equations by applying elementary row operations. When we apply elementary row operations, there
MAT 1341: DETERMINANTS II
1. Properties of determinants
One can use elementary operations to simplify the computation of the determinant:
Theorem 1.1. Let A be an n n matrix.
(1) If B is obtained by interchanging any two rows of A, then
det(B) = det(A)
(2