Find a standard basis vector for R^3 that can be added to the set {𝐯1, 𝐯2} to produce a basis fo

step 1 Recall that our stand -dus basis vectors are E1, E2, and E3. So in this problem, we want to see

Find a standard basis vector for R^3 that can be added to...

Find a standard basis vector for $R^{3}$ that can be added to the set $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ to produce a basis for $R^{3}$
(a) $\mathbf{v}_{1}=(-1,2,3), \mathbf{v}_{2}=(1,-2,-2)$
(b) $\mathbf{v}_{1}=(1,-1,0), \mathbf{v}_{2}=(3,1,-2)$

Key Concepts

Linear Independence

Linear independence is the property of a set of vectors where no vector in the set can be written as a linear combination of the others. When extending a set of vectors to form a basis, any new vector added must not be expressible in terms of the existing ones, ensuring that the full set remains linearly independent. This is crucial to forming a valid basis for the vector space.

Basis

A basis of a vector space is a set of vectors that is both linearly independent and spans the entire space. In the context of R³, a basis consists of three vectors such that any vector in R³ can be written as a linear combination of these three vectors. For a subset of vectors to be extended to a full basis, any additional vector must not lie in the span of the current set.

Standard Basis

The standard basis of R³ is the set of unit vectors along the coordinate axes, usually denoted as e? = (1, 0, 0), e? = (0, 1, 0), and e? = (0, 0, 1). These vectors are mutually orthogonal, linearly independent, and collectively span R³. When the problem asks for a standard basis vector, it means choosing one of these vectors that is not already in the span of the given set.

Transcript

00:01 Recall that our stand -dus basis vectors are e1, e2, and e3.

00:05 So in this problem, we want to see which of these vectors can be added to a set of vectors to show that this new set is a basis.

00:12 Now, since all of these vectors are of length 3 and we're in r3, theorem 4 .5 .4 tells us that we just need to check that the set is linearly independent.

00:23 So let's look at our first two vectors, and i put those vectors into a matrix.

00:29 Now we see if we add these two vectors together, we just get e3.

00:34 So we know our answer e3 is not linearly independent of these two.

00:39 So let's go ahead and see if e2 is linearly independent.

00:44 So we put e2 into the last column of our matrix, and then we're going to try to reduce to row echelon form.

00:52 So first we'll use the first row to cancel out the second row, and we'll do that by multiplying the first row by two and adding it to the second.

01:02 So that gives a 0, a 0, and a 1.

01:07 And then to cancel out the 3 in our third row, we're going to multiply the top row by 3, and we will add it to the third row.

01:15 So that gives us a 0, that gives us a 1, and that gives us a 0.

01:22 So now we can use the bottom row to cancel out the 1 in the second entry of the top row, giving us the following.

01:34 And all i'm doing here is just taking negative of the last row and adding it to the first row.

01:50 And now we can see that each row in each column only has one entry, which implies these three vectors are linearly independent.

01:58 So the answer to part one is e2.

02:01 Now let's go ahead and put the vectors from part two into a matrix...

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