W1 W2 Linear Algebra OSU

What is a linear combination?

A sum of the form a₁x₁ + a₂x₂ + ... + aₙxₙ with aᵢ in a field (usually ℝ)

What makes a linear equation homogeneous?

When the constant term b = 0.

What is a linear system?

A collection of linear equations with the same variables

What does it mean for two systems to be equivalent?

They have the same solution set

What does it mean for a system to be dependent?

It has infinitely many solutions.

hat is the preferred way to write solutions to linear systems?

In column-vector notation, especially for dependent solutions.

ex:

[4/3] + s[-1/3] + t[1]

How is the solution to a consistent system related to its homogeneous counterpart?

It's a particular solution plus all solutions to the homogeneous system.

What type of result do you get when you multiply an m×n matrix by an n×1 column vector?

An m×1 column vector

What property is illustrated by A(B + C) = AB + AC?

Distributive property of matrix multiplication.

What does the matrix product Ax represent?

A linear combination of the columns of A using components of x as coefficients.

How do you write a system of equations as a matrix equation?

As Ax=b where A contains coefficients, x contains variables, and b is the result vector.

What is the span of vectors v1,v2,...,vk

All linear combinations of those vectors.

How can you check if b∈(v1,...,vk)?

Form and row reduce the augmented matrix [v1 v2 ... vk ∣ b] to see if it's consistent.

How many solutions can a linear system have?

None, exactly one, or infinitely many

What kind of geometric object is the span of one nonzero vector? Two?

One: a line through the origin. Two: a line or a plane through the origin.

What is (AB)ij in matrix multiplication?

It's the dot product of the i-th row of A and the j-th column of B.

What does matrix multiplication distribute over?

Addition:

A(B+C)=AB+AC,

(B+C)A=BA+CA

Is matrix multiplication commutative?

No, AB≠BA in general

What is the transpose of a matrix?

A matrix where rows and columns are swapped:

(A^T)_ij= a_ji

What is a symmetric matrix?

A square matrix where A=A^T

What do we call a matrix that is not invertible?

Singular

What is the determinant of a 2×2 matrix

ad−bc

What is the inverse of a 2x2 matrix?

1/ (ad - bc) * [ d -b ]

[ -c a ]

What does x=A−1b represent?

The unique solution to the linear system Ax=b, if A is invertible

Is the inverse of a matrix unique

yes

What is (AB)^-1?

B^-1 * A^-1

What is (A^T)^-1 ?

(A^-1)^T