TAMU Math 304 Final Flashcards

Questions and Answers

What kind of matrix can be factored into elementary matrices?

Only invertible ones

What is the associative property for matrices?

(AB)C = A(BC)

What is the distributive property for determinants?

det(AB) = det(A) * det(B)

What is a similar matrix?

When two matrices A, B and another invertible matrix P satisfy B = P⁻¹AP Signup and view all the answers

What is a diagonalizable matrix?

A matrix that can be written in factored form as PDP⁻¹, where D is a diagonal matrix and P is an invertible matrix Signup and view all the answers

What is the characteristic polynomial?

det(A - λI) Signup and view all the answers

When is a matrix not invertible?

If det(A) = 0, A is not invertible. Signup and view all the answers

What does dim(N(A)) represent?

Number of null basis elements Signup and view all the answers

What is the change of basis formula?

B = V⁻¹AV Signup and view all the answers

What is the process for finding a least squares solution?

<ol> <li>Find AᵀA 2. Find Aᵀb 3. Solve AᵀAx = Aᵀb for x</li> </ol> Signup and view all the answers

What is the Gram-Schmidt Process?

<ol> <li>u₁ = v₁ 2. u₂ = v₂ - proj(u₁, v₂) 3. u₃ = v₃ - proj(u₁, v₃) - proj(u₂, v₃)</li> </ol> Signup and view all the answers

How do you calculate the dot product of vectors u and v?

uᵀv Signup and view all the answers

When is vector u orthogonal to vector v?

uᵀv = 0 Signup and view all the answers

What are the criteria for an orthonormal set {u₁, u₂, u₃}?

<ol> <li>= 1 2. = 0</li> </ol> Signup and view all the answers

What is a singular matrix?

A singular matrix is a square matrix with no inverse. Its determinant is zero. Signup and view all the answers

What is the inverse of the product of two matrices (AB)⁻¹?

B⁻¹A⁻¹ Signup and view all the answers

If {x₁, x₂, x₃} is a basis for R³, what is another basis for R³?

{x₁, x₂, x₂ + x₃} Signup and view all the answers

What is L(ax + By)?

aL(x) + bL(y) Signup and view all the answers

What is the subspace criterion?

<ol> <li>S contains 0 vector 2. For any two vectors u and v in S, u + v is in S 3. For any vector u in S, a*u is also in S.</li> </ol> Signup and view all the answers

What does it mean if the determinant of two vectors is nonzero?

They are LINEARLY INDEPENDENT Signup and view all the answers

How do you determine linear independence?

<ol> <li>Convert to row echelon form 2. The number of linearly independent vectors is equal to the dimension.</li> </ol> Signup and view all the answers

What are symmetric matrix properties?

<ol> <li>Sum of symmetric matrices is symmetric 2. The same is not true for products 3. A^n is symmetric if A is symmetric 4. A⁻¹ is only symmetric if A is symmetric.</li> </ol> Signup and view all the answers

What is the dimension of span()?

Dimension will be less than the number of maximum vectors that are linearly dependent, and greater than or equal to the minimum number of linearly dependent vectors. Signup and view all the answers

How can you check if a vector V is in the column space of a matrix A?

Try to solve Ax = V Signup and view all the answers

Study Notes

Matrix Properties and Definitions

Only invertible matrices can be factored into elementary matrices.
The associative property of matrices states that (AB)C = A(BC).
Determinants follow the distributive property: det(AB) = det(A) * det(B).
A matrix B is similar to matrix A if there exists an invertible matrix P such that B = P⁻¹AP.
Diagonalizable matrices can be expressed as PDP⁻¹, where D is diagonal and P is invertible.

Determinants and Invertibility

The characteristic polynomial of a matrix A is given by det(A - λI).
A matrix A is not invertible if det(A) = 0.

Dimensions and Bases

The dimension of the null space, dim(N(A)), equals the number of null basis elements.
The change of basis formula is expressed as B = V⁻¹AV.
In R³, if {x1, x2, x3} forms a basis, then {x1, x2, x2 + x3} also forms a basis.

Least Squares and Orthogonality

For the least squares solution, compute AᵀA, then Aᵀb, and solve AᵀAx = Aᵀb for x.
The Gram-Schmidt Process is used to orthogonalize vectors, starting with u₁ = v₁ and adjusting each subsequent vector by projecting onto previous ones.
The dot product of vectors u and v is denoted as uᵀv, and u is orthogonal to v if uᵀv = 0.

Orthogonal and Orthonormal Sets

A set {u₁, u₂, u₃} is orthonormal if it satisfies two conditions: the norm of each vector equals 1, and they are pairwise orthogonal (dot products equal 0).

Singular and Linear Independence

A singular matrix has no inverse, indicated by a determinant of zero.
The inverse of the product of two matrices is given by (AB)⁻¹ = B⁻¹A⁻¹.
To determine linear independence, convert a set of vectors to row echelon form; the number of linearly independent vectors corresponds to the dimension of the space.

Symmetric Matrices

Symmetric matrices have specific properties:
- The sum of symmetric matrices is symmetric.
- The product of symmetric matrices may not be symmetric.
- Powers of symmetric matrices (A^n) remain symmetric if A is symmetric.
- The inverse of a symmetric matrix (A⁻¹) is symmetric only if A is symmetric.

Span and Subspaces

The dimension of span(S) is less than the number of maximum vectors if they are linearly dependent, and at least equal to the minimum number of linearly independent vectors.
A subset S qualifies as a subspace if it contains the zero vector, is closed under addition, and is closed under scalar multiplication.

Column Space

To determine if vector V is in the column space of matrix A, attempt to solve the equation Ax = V.

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Test your knowledge with these flashcards covering key concepts from TAMU Math 304. The flashcards summarize important properties and definitions related to matrices and determinants, helping you prepare effectively for the final exam. Ideal for students seeking a quick review of their understanding in matrix algebra.