Find bases for the image of ( 1 2 -1 0 3 -3 2 -4 6

Find bases for the image of ( 1 2 -1 0 ...

Key Concepts

Basis

A basis of a subspace is a set of vectors that are both linearly independent and span the entire subspace. This means every vector in the subspace can be uniquely written as a linear combination of the basis vectors. The basis provides a minimal and efficient coordinate system for the subspace, and finding a basis is a fundamental task in linear algebra.

Image of a Linear Transformation (Column Space)

The image of a linear transformation, often referred to as the column space in the context of matrices, is the set of all possible output vectors obtained by applying the transformation. In practical terms, it is the set of all linear combinations of the columns of the matrix. Understanding this concept is essential for figuring out what subspace a matrix maps to and determining its dimension (rank).

Row Reduction (Gaussian Elimination)

Row reduction is a systematic method for simplifying a matrix into a row echelon form or reduced row echelon form by applying elementary row operations. This process helps identify the pivot positions, which in turn reveals the linearly independent columns that form a basis for the column space. It is one of the primary methods for determining the rank and basis of a matrix.

Linear Independence

Linear independence is a property of a set of vectors indicating that no vector in the set can be written as a linear combination of the others. This property is essential for a set of vectors to qualify as a basis; all basis vectors must be free from redundancy, ensuring that each vector contributes a unique direction in the vector space.

Change of Basis

Change of basis refers to the process of expressing a set of basis vectors in terms of another set of basis vectors for the same subspace. Demonstrating how one basis can be written as a linear combination of another is a key step in verifying that two different sets of vectors indeed span the same subspace, thereby confirming that both are valid bases.

Transcript

00:01 So for this question, they want us to determine a basis for the subspace of square two by two matrices with all entries.

00:13 Okay, it's going to be a subspace of it, spanned by this matrix.

00:22 13, negative 1 ,000, 0 ,000, okay, negative 1, 411, and 5, negative 6 ,000 ,000, and 5, negative 6, negative, negative 5 .1.

00:38 Okay, now why do they left? well, what are spans? any matrices that can be written as a linear combination of these.

00:45 In other words, the collection of matrices, where you take some coefficient, multiply it by this, choose another coefficient, multiply it by this, another, another, multiply them all, and add them all together.

00:58 Well, what happens when you multiply zero by anything, zero, zero, zero, zero, the zero vector by anything? in this case, zero matrixy.

01:04 Well, you just get the zero matrixy again, so it's adding absolutely nothing to our span.

01:10 Okay.

01:12 Well, next thing, i don't know if these are independent from looking at them, because first off, they're in matrices form, but they're completely described by their coefficients, so we can describe them as vectors in our 4.

01:27 Okay, so we can ask if those are linearly independent by seeing the solutions to the linear system of them for zero.

01:37 If they're a basis, there's only one way to get the zero vector, or the zero matrix.

01:44 Okay, in terms of choices of scalers for your vectors.

01:50 Let's write them out.

01:50 There are one, three, negative one, two.

01:54 So i unrolled this matrix by rows into a vector.

01:58 Same thing for the second one, negative one, four, one, one.

02:03 I have to keep consistent.

02:06 Five, negative six, negative five, 1.

02:10 Okay.

02:12 So maybe they're all linearly independent.

02:18 I'll be able to see that if i do a little gaussian elimination.