Math 3203 (Fall 2026)

Section 1.7-Linear Independence

Defintion: An indexed set of vecotrs {v1, v2,…, vp} in R^n is said to be linearly independent if the vector equation x1v1 +…+ xpvp=0 has only the trivial solution x1=x2=…=xp=0. The set is linearly dependent if there exist weights c1…cp not all zero such that c1v1 +…+ cpvp=0.

Suppose we have a matrix A = [a1,…an]. The matrix equation Ax=0 can be written as vector equation x1a1 +x2 a2 +…+ xnan = 0. Each linearly dependent relaiton among the columns of A corresponds to a nontrivial solution of Ax=0. The columns of Matrix A are linearly independent if and only if the equation Ax=0 has ONLY the trivial solution

Sets of one or two vectors

A set combining only one vector v is linearly independent c1,v1=0 if and only if v is NOT the zero vector

Determine if the set of vectors {[3 1], [6 2]} is linearly independent. Does x1[3 1] + x3[6 2] = [0 0] have a nontrivial solution?

Yes, x1 = -2 and x2 = 1. So the set of vectors is linearly dependent. A set of two vectors {v1, v2} is linearly dependent if at least one of the vectors is a scalar multiple of the other. The set {v1, v2} is linearly independent if and only if neither of the vectors is a scalar multiple of the other.

Geometrically: Two vectors are linearly dependent if and only if they’re on the same line through the origin

Theorem 1.7-Characteristics of linearly dependent sets

An indexed set S = {v1,…,vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact if S is linearly dependent and v ≠ 0 then teh same vj (with j>1) is a linear combination of the preceding vecors v1,…vj-1)

NOTE: This does not say that every vector in a linearly dependent set is a linear combination of the preceding vectors.

In general, if u, v, w, are in R³ and u & v are linearly independent thatn {v, v, w} is linearly dependent if and only if w is in the plane spanned by u & v.

Theorem 1.8: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v1,…vp} in R^n is linearly dependent if p>n.

Proof: Let p>n. Let v12,…vp be vectors in R^n.

Let A=[v1,…vp]. The A is an nxp matrix. THe equaiton Ax=0 corresponds to a system of n equaions in p. Since p>n, there are more column then rows. Hence Ax=0 has a nontrivial solution. Thus the columns A are linearly dependent. Hence the set {v1,…, vp} is linearly dependent

Theorem 1.9: If a set S={v1,…vp} in R^n contains teh zero vector then teh set is linearly dependent.

Proof: Let v1,…vp be vectors in R^n. Let S={v1,…vp}. Suppose one of these vectors is the zero vector. Remember the vectors so that v1=0. THen 1v1 + 0v2 + 0v3 +…+ 0vp = 0. THus S is linearly dependent


Section 1.8 - Introductions to Linear Transformation

A transformation or function or mapping T from R^n to R^m is a rule that assigns to each vector x in R^n a vector T(x) in R^m.The set R^n is the domain of T. The set R^m is the codomain of T. The notation T:R^n→R^m indicates that the domain of T is R^n and the codmain is R^m. For x in R^n, the vector T(x) in R^m is called teh image of x. The set of all images T(x) is called teh range of T.

Let A be an mxn matrix. Define T(x) by T(x)=Ax. Then the transformation T has Domain R^n and codomain R^m, so TR^n→R^m. If A=[a1,…,an] and x = x1,…xn], then T(x)=Ax. T(x)=Ax=x1a1+…+xnan.

The range of T is the set of all linear combination of the column of A. The range of T is equal to the span o fthe columns of A.

Define: A transformation or mapping T is linear if

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 all u in the domain of T

Linear transformation preserved teh operation sof vectors addition and scalar multiplication

From Section 1.4: Theorem 1.5- If A is an mxn matrix, u+v are vectors in R^n and c is a scalar, then A(u+v)=Au+Av and A(cu)=c(Au)

Let A be an mxn matrix and define T:R^n→R^m by T(x)=Ax then theorem 1.5 tells us that T is a linear transformation.

Definition: Given a scalar r >= 0, define T:R²→R² by T(x)=rx. The transformation T is called a contraction when 0<=r<=1 and dilation when r>1

Section 1.9 - The Matrix of a Linear Transformation

The columns of I2 are e1=[1 0] and e2 = [0 1]

Theorem 1.10: Let T: R^n → R^m be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax for all x in R^n. In fact, A is the mxn matrix whose jth column is the vector T(ej), where ej is the jth column of the identity matrix in R^n. A = [T(e1) T(e2)…T(en)]

The matrix A alone is called the Standard Matrix for the Linear Transformation T

Definiton: A mapping T:R^n → R^m is said to be onto R^m if each b in R^m is the image of at least one x in R^n. T is onto R^m when the range (T) = R^m = codomain(T). The mapping T is NOT onto R^m when there is some b in R^n for which the equation T(x)=b has no solution

Definition: A mapping T:R^n → R^m is said to be one to one (1-1) if each b in R^m is the image of at most one x in R^n. T is 1-1 for each b in R^m, the equation T(x)=b has either a unique solution or no solution.

The mapping T is NOT 1-1 such that some b in R^m is the image of more than one vector in R^n

Theorem 1.11: Let T:R^n→ R^n be a linear transformation. Then T is 1-1 if an donly if the equation T(x)=0 ahs only the trivial solution.

Theorem 1.12: Let T:R^n→R^m be a linear transformation and Let A be the standard matrix for T. Then:

a) T maps R^n onto R^m if and only if the caolumn of A span R^m (which happens precisely when A has a pivot in each row)

b) T is 1-1 if and only if the columns of A are linearly independent (which happens precisely when A has a pivot in each column)

Section 2.1 - Matrix Operations

If A is an mxn matrix, then the scalar entry in the ith row and jth column of A is denoted aij and is called teh (ij)-entry of A

Theorem 2.1:

Let A, B and C, be matrices of teh same size, and r and s be scalars

  1. A + B = B + A

  2. (A + B) + C = A + (B + C)

  3. A + 0 = A

  4. r(A+B) = rA + rB

  5. (r+s)A = rA + sA

  6. r(sA) = (rs)A

When multiplying matrices let’s say A is 3×2 and matrix B is 2×2. If the columns of A matches rows of B the product is defined. (3×2 2×2). The row size of A and the column size of B tells you the size of the product. (3×2 2×2)

Properties of Matrix Multiplication. Theorem 2.2

Let A be an mxn matrix, and let B and C have sizes for which the indicated sums and products are defined.

  1. A(BC) = (AB)C

  2. A(B+C) = AB + AC

  3. (B+C)A = BA + CA

  4. For any scalar r, r(AB) = (rA)B = A(rB)

  5. InA = A = AIn

Warnings:

  1. In general, AB ≠ BA

  2. If AB = AC, it is not true in general that B=C

  3. If AB=0, you can’t conclude in general that either A = 0 or B = 0

If A is an nxn matrix and k is a positive integer, then A^k denotes the product of k copies of A.

A^k = A…A. And A^0 is the indenity matrix

Definition: Given and mxn matrix A, the transpose of A is the nxm matrix, denoted A^T, where columsn are formed from the corresponding rows of A

Theorem 2.3: Let A and B denote matrices whose sides are appropriate for the following sums and products

  1. (A^T)^T = A

  2. (A+B)^T = A^T + B^T

  3. For any scalar r, (rA)^T = rA^T

  4. (AB)^T = B^TA^T

Section 2.2 - The Inverse of a Matrix

Definition: An nxn matrix A is said to be invertible if there is an nxn matrix C such that CA=I and AC=I

In this case, C is an inverse of A. In fact, C is unique and this unique inverse is denoted A^-1, So AA^-1=I and A^-1A = I

A amtrix that is NOT invertible is also called a singular matrix, and an invertible matrix is called a nonsingular matrix

Special Case: 2×2 amtr5ices

Definition: If A is a 2×2 matrix with A = [a b c d], then the quantity ad-bc is called the determinant of A and we write detA = ad-bc

Theorem 2.4: Let A = [a b c d]. IF detA≠0, the A is invertible and A^-1= 1/(ad-bc)[d -a -c b]. If detA=0, the A is not invertible

Theorem 2.5: If A is an invertible nxn matrix, then for each b in R^n, the equation Ax=b ahs the unique solution x=A^-1b

Proof: Let A be and invertible matrix. Let b be in R^n. Note that A(A^-1b) = AA^-1b=Ib=b, so x=A^-1b is a solution to the equation Ax=b

Theorem 2.6:

a) If A is an invertible amtrix, then A^-1 is invertible (A^-1)^-1=A

b) If A and B are nxn invertible matrices then AB, and (AB)^-1=B^-1A^-1

This part(b) generalizes to:

The product of mxn invertible matrices is invertible , and the inverse the product iverses in their reverse order

c) If A is an invertible matrix, then so is A^T, and (A^T)^-1 = (A^-1)^T

Theorem 2.7: An nxn matrix A is invertible if and only if A is row equivalent to In and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A^-1

Algorithm fo r Finding A^-1

Row reduce teh augmented matrix [A|In]. If A is row equivalent to In then [A|In] is row equivalent to [In|A^-1]. Otherwise A doesn’t have an inverse

Section 2.3 - Characterization of Invertible Matrices

Theorem 2.8 The Invertible Matrix Theorem (Memorize at least 6 of these not including part 1)

  1. A is invertible

  2. A is row equivalent to the nxn identity matrix

  3. A has n pivot position

  4. The equation Ax=0 has only trivial solution (aka no free variable)

  5. The column of A form a linearly independent set

  6. The linear transofrmation x→Ax is 1-1

  7. The equation Ax=b has a unique solution for each b in R^n

  8. The column of A span R^n

  9. The linear transformation x → Ax maps R^n onto R^n

  10. There is an nxn matrix C such that CA = I

  11. There is an nxn matrix D such that AD=I

  12. A^T is an invertible matrix

  13. Either all of theses are true or all are false

The theorem ONLY applies to square matrices. Let A and B be square matrices, If AB=I, then A and B are both invertible with B=A^-1 and A=B^-1