Linear Algebra Midterm Review Flashcards

This set of flashcards covers the key concepts, procedures, and notation necessary for studying Linear Algebra, specifically for the Midterm exam.

How do I find the Null Space of a matrix?

1. Solve the equation Ax = 0. 2. Write the solutions in vector form. 3. Identify vectors corresponding to free variables as the basis.

What is the procedure to determine if a set of vectors is linearly independent?

1. Form a matrix with the vectors as columns. 2. Transform the matrix to RREF. 3. If there's a pivot in every column, the vectors are independent; if free variables exist, they are dependent.

How do I write the parametric form of a solution for a system with infinite solutions?

Solve for pivot variables in terms of free variables, and express it as S = { xparticular + u * vhomogeneous : u is a real number }.

Steps to find the Column Space Basis (im A) of a matrix:

1. Reduce the matrix to RREF. 2. Identify pivot columns. 3. Select corresponding columns from the ORIGINAL MATRIX.

What differentiates a pivot in RREF from a free variable?

A pivot is the leading 1 in a column of RREF, indicating a specific solution value, while a free variable corresponds to columns without pivots and can take on any value.

True or False: I use RREF columns for the Column Space basis.

False. Use the corresponding columns from the ORIGINAL MATRIX to form the Column Space basis.

What is the method for Polynomial Interpolation?

Set up a system using given points in the form of an augmented matrix [1, x, x^2 | y]. Solve for vector coefficients.

How do you solve an Integer Optimization Problem?

1. Form parametric equations from the system. 2. Determine constraints for non-negative variables. 3. Find integer solutions within the range and calculate costs.

How do you find coordinates of vector x relative to basis B?

Solve the augmented matrix [v1 v2 | x] where v1 and v2 are the basis vectors.

What are the RREF rules for a matrix?

1. Leading 1s must be to the right of leading 1s in rows above. 2. Leading 1s must be the only non-zero entries in their columns. 3. All-zero rows must be at the bottom.

How do I find the Null Space of a matrix?

1. Solve the equation A\mathbf{x} = \mathbf{0}. 2. Write the solutions in vector form. 3. Identify vectors corresponding to free variables as the basis.

What is the procedure to determine if a set of vectors is linearly independent?

1. Form a matrix with the vectors as columns. 2. Transform the matrix to RREF. 3. If there's a pivot in every column, the vectors are independent; if free variables exist, they are dependent.

How do I write the parametric form of a solution for a system with infinite solutions?

Solve for pivot variables in terms of free variables, and express it as S = { \mathbf{x}{particular} + u \cdot \mathbf{v}{homogeneous} : u \in \mathbb{R} }.

Steps to find the Column Space Basis (im A) of a matrix:

1. Reduce the matrix to RREF. 2. Identify pivot columns. 3. Select corresponding columns from the ORIGINAL MATRIX.

What differentiates a pivot in RREF from a free variable?

A pivot is the leading 1 in a column of RREF, indicating a specific solution value, while a free variable corresponds to columns without pivots and can take on any value.

True or False: I use RREF columns for the Column Space basis.

False. Use the corresponding columns from the ORIGINAL MATRIX to form the Column Space basis.

What is the method for Polynomial Interpolation?

Set up a system using given points in the form of an augmented matrix [1, x, x^2 \mid y]. Solve for vector coefficients.

How do you solve an Integer Optimization Problem?

1. Form parametric equations from the system. 2. Determine constraints for non-negative variables. 3. Find integer solutions within the range and calculate costs.

How do you find coordinates of vector x relative to basis B?

Solve the augmented matrix [\mathbf{v}1 \mathbf{v}2 \mid \mathbf{x}] where \mathbf{v}1 and \mathbf{v}2 are the basis vectors.

What are the RREF rules for a matrix?

1. Leading 1s must be to the right of leading 1s in rows above. 2. Leading 1s must be the only non-zero entries in their columns. 3. All-zero rows must be at the bottom.

What is the Rank-Nullity Theorem?

For an m \times n matrix A, the rank of the matrix plus the nullity of the matrix equals the number of columns: rank(A) + nullity(A) = n.

How do you find the basis for the Row Space (Row A)?

1. Transform the matrix to RREF. 2. The non-zero rows of the RREF matrix form the basis for the Row Space.

What are the three criteria for a subset H to be a subspace of \mathbb{R}^n?

1. The zero vector \mathbf{0} is in H. 2. H is closed under vector addition. 3. H is closed under scalar multiplication.

How do you calculate the inverse of a 2 \times 2 matrix A = \begin{bmatrix} a & b \ c & d \end{bmatrix}?

If ad - bc \neq 0, then A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}.

What is the standard matrix of a linear transformation T: \mathbb{R}^n \to \mathbb{R}^m?

The matrix A = [T(\mathbf{e}1) \ T(\mathbf{e}2) \ \dots \ T(\mathbf{e}n)], where \mathbf{e}i are the columns of the identity matrix I_n.

What defines a transformation as 'Onto' (surjective)?

A transformation T: \mathbb{R}^n \to \mathbb{R}^m is onto if the columns of its standard matrix span \mathbb{R}^m, meaning there is a pivot in every row.

What defines a transformation as 'One-to-One' (injective)?

A transformation T: \mathbb{R}^n \to \mathbb{R}^m is one-to-one if the equation A\mathbf{x} = \mathbf{0} has only the trivial solution, meaning there is a pivot in every column (no free variables).