Purdue MA 265 Flashcards

Questions and Answers

What are the two conditions for a square matrix to be singular?

Its RREF contains a zero row and its det = 0.

What is an elementary matrix?

A matrix which differs from the identity matrix by one single elementary row operation.

All elementary matrices are non-singular.

True

If A is a matrix, then what is the determinant of A transpose?

The same as det(A). Signup and view all the answers

How does interchanging rows affect det(A)?

det(A) => -det(A). Signup and view all the answers

How does multiplying a scalar k to a row affect det(A)?

det(A) => k * det(A). Signup and view all the answers

What happens to det(A) if we multiply a scalar k to a row and add it to another row?

det(A) remains the same. Signup and view all the answers

What does it mean for matrices A and B to be row equivalent?

B can be obtained by applying a finite number of elementary row operations to A. Signup and view all the answers

If E is an elementary matrix, then det(EA) = ...

det(E) * det(A). Signup and view all the answers

If matrix A is non-singular, what can be said about det(A)?

det(A) cannot equal zero. Signup and view all the answers

What is the relationship between det(AB) and the determinants of A and B?

det(AB) = det(A) * det(B). Signup and view all the answers

If det(A) does not equal zero, is matrix A linearly independent or dependent?

Linearly independent. Signup and view all the answers

What is a symmetric matrix?

A^T = A. Signup and view all the answers

What is a skew symmetric matrix?

A^T = -A. Signup and view all the answers

What is Cramer’s rule?

Cramer’s rule relates the solutions of a system of linear equations to determinants. Signup and view all the answers

What are the properties of matrix addition?

<ol> <li>A + B = B + A; 2. A + (B + C) = (A + B) + C; 3. There is a unique mxn matrix 0 such that A + 0 = A for every A; 4. For every mxn matrix A, there exists -A such that A + -A = 0.</li> </ol> Signup and view all the answers

What are the properties of matrix multiplication?

<ol> <li>A(BC) = (AB)C; 2. (A + B)C = AC + BC; 3. C(A + B) = CA + CB.</li> </ol> Signup and view all the answers

What are the properties of the transpose of a matrix?

<ol> <li>(A^T)^T = A; 2. (A + B)^T = A^T + B^T; 3. (AB)^T = B^T * A^T; 4. (rA)^T = rA^T.</li> </ol> Signup and view all the answers

What are three differences between matrix multiplication and multiplication of real numbers?

<ol> <li>AB need not equal BA; 2. AB may be 0 with A not equal to 0 and B not equal to 0; 3. AB may equal AC with B not equal to C.</li> </ol> Signup and view all the answers

If A and B are nonsingular nxn matrices, then what can be said about AB?

AB is nonsingular and (AB)^-1 = B^-1 * A^-1. Signup and view all the answers

If A is nonsingular, what can be said about the transpose of A and the inverse of A^T?

A^T is nonsingular and (A^-1)^T = (A^T)^-1. Signup and view all the answers

If A is a matrix, then how do you express the determinant of A?

det(A) = det(A^T). Signup and view all the answers

What is the relationship between det(A^-1) and det(A)?

det(A^-1) = 1/(det(A)). Signup and view all the answers

What defines a subspace W of a vector space V?

It is closed for addition and scalar multiplication. Signup and view all the answers

What can be said about the span of a set of vectors S in V?

The set of all vectors in V that are linear combinations of the vectors in S and span S is a subspace of V. Signup and view all the answers

When are vectors V1, ..., Vr in a vector space said to be linearly dependent?

There exist constants a1, a2, ..., ar, not all zero, such that a1V1 + a2V2 + ... + akVk = 0. Signup and view all the answers

What condition indicates that a set S of n vectors in Rn is linearly independent?

det(A) does not equal 0. Signup and view all the answers

If dimV = n, what can be said about the vectors in V?

<ol> <li>Any subset of m > n vectors must be linearly dependent; 2. Any subset of m < n vectors cannot span V.</li> </ol> Signup and view all the answers

What can be said about a linearly independent set of vectors S in a finite-dimensional vector space V?

There is a basis T for V that contains S. Signup and view all the answers

If S is a linearly independent set of vectors in an n-dimensional vector space V, what can be concluded?

S is a basis. Signup and view all the answers

If S spans V, what can be inferred?

S is a basis for V. Signup and view all the answers

What does it mean if spanS = V for a finite subset S of vector space V?

A maximal independent subset T of S is a basis for V. Signup and view all the answers

How can we find the basis for V if V = { x | Ax = 0 , A exists in Mmn } is a subspace for Rn?

<ol> <li>Start by creating the augmented matrix [A|0]; 2. Transform to RREF, [B|0], where B has r nonzero rows.</li> </ol> Signup and view all the answers

Study Notes

Matrix Definitions and Properties

A square matrix is singular if its reduced row echelon form (rref) contains a zero row or its determinant equals zero.
An elementary matrix results from applying a single elementary row operation to the identity matrix.

Determinant Properties

All elementary matrices are non-singular, meaning they have a non-zero determinant.
The determinant of a matrix remains unchanged when transposed.
Interchanging two rows of a matrix negates its determinant (det(A) becomes -det(A)).
Multiplying a row by a scalar ( k ) scales the determinant by ( k ) (det(A) becomes k * det(A)).
Adding a multiple of one row to another row leaves the determinant unchanged.
If ( A ) and ( B ) are matrices, then det(AB) = det(A) * det(B).
If det(A) is non-zero, matrix ( A ) is linearly independent.

Matrix Types

A symmetric matrix satisfies the property ( A^T = A ).
A skew-symmetric matrix satisfies ( A^T = -A ).

Cramer's Rule

Used to solve systems of linear equations with a non-zero determinant.

Properties of Matrix Operations

For matrix addition:
- A + B = B + A (commutative),
- A + (B + C) = (A + B) + C (associative),
- There exists a zero matrix such that A + 0 = A,
- Each matrix has an additive inverse where A + (-A) = 0.
For matrix multiplication:
- A(BC) = (AB)C (associative),
- (A + B)C = AC + BC (distributive),
- C(A + B) = CA + CB (distributive).

Properties of Transpose

A double transpose returns the original matrix: ( (A^T)^T = A ).
The transpose of a sum of matrices equals the sum of their transposes.
The transpose of a product reverses the order: ( (AB)^T = B^T A^T ).
The transpose of a scalar multiplied by a matrix is the scalar multiplied by the transpose of the matrix: ( (rA)^T = rA^T ).

Linear Independence and Basis

Vectors ( V_1, \ldots, V_r ) are linearly dependent if there are constants ( a_1, a_2, \ldots, a_r ) (not all zero) such that ( a_1V_1 + a_2V_2 + \cdots + a_rV_r = 0 ).
A set of ( n ) vectors in ( \mathbb{R}^n ) is linearly independent if the determinant of their corresponding matrix ( A ) is non-zero.
In an ( n )-dimensional vector space ( V ):
- Any subset of more than ( n ) vectors must be linearly dependent.
- Any subset with fewer than ( n ) vectors cannot span ( V ).
A finite set ( S ) of vectors in ( V ) with span equal to ( V ) has a maximal independent subset ( T ) that forms a basis for ( V ).
If ( S ) spans ( V ) or is a linearly independent set, it is a basis for ( V ).

Finding the Basis

To find the basis for ( V ) defined by ( V = {x | Ax = 0} ):
- Create the augmented matrix [A|0],
- Transform it into reduced row echelon form (rref) to identify non-zero rows, which provide the basis.

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Test your knowledge of key concepts in linear algebra with these flashcards from Purdue's MA 265 course. Topics include singular matrices, elementary matrices, and determinants. Perfect for quick review and exam preparation.