Linear Algebra 1

Theorem 1.18: when is a linear system consistent?

Exactly when the last column of the augmented matrix is NOT a pivot column

A system is consistent and has at least one free variable. How many solutions does it have?

Infinitely many

Theorem 1.36: which statement is equivalent to ‘Ax= b has a solution for EVERY b in ℝᵐ"‘

A has a pivot in every ROW

The columns of A are linearly independent exactly when…

There is a pivot point in every column

Proposition 1.43: Ax = 0 has a nontrivial solution if and only if…

There is at least one free variable

Theorem 1.46: if p is one solution of Ax = b, then every solution of Ax = b has the form…

p + v, where Av = 0 (only when Ax = b is consistent)

Theorem 1.53: p vectors in ℝⁿ with p > n are…

always linearly dependent

If {v₁, …, vₚ} is linearly dependent (p ≥ 2), which is GUARANTEED?

At least one vector is a linear combination of the others

For T(x) = Ax where A is m×n, the RANGE of T is…

the span of the columns of A (a subset of ℝᵐ)

Theorem 1.62: the standard matrix of a linear T: ℝⁿ → ℝᵐ is…

[T(e₁) T(e₂) … T(eₙ)], an m×n matrix

What is the fastest way to show T(x₁, x₂) = (x₁ + 3, x₂) is NOT linear?

T(0, 0) = (3, 0) ≠ (0, 0). Every linear transformation satisfies T(0) = 0. But remember: T(0) = 0 is necessary, not sufficient (e.g., |x₂| breaks linearity even though T(0) = 0).

When is the matrix-vector product Ax defined?

When the number of columns of A equals the number of entries of x

Which statement is ALWAYS true for any vectors u and v in ℝⁿ?

The zero vector is in span{u, v}

A is n×n and has a pivot in every row. What else must be true?

It has a pivot in every column, its RREF is Iₙ, and its columns are independent

A is a 4×3 matrix. Which statement MUST be true?

Ax = b is inconsistent for some b in ℝ⁴

What does RREF require beyond REF?

Every pivot equals 1, and every entry above a pivot is 0

Which is NOT a legal elementary row operation?

Ri ← 0·Ri

Scaling must use a NONZERO constant; multiplying by 0 destroys information. The three legal operations: replacement, interchange, scaling by c ≠ 0.

Theorem 1.33: which of these has the same solution set as Ax = b, where A = [a₁ … aₙ]?

x₁a₁ + … + xₙaₙ = b, and the system with augmented matrix [a₁ … aₙ b]