Verify that the vectors 𝐯1=(-(3)/(5), (4)/(5), 0), 𝐯2=((4)/(5), (3)/(5), 0), 𝐯3=(0,0,1) form an

Verify that the vectors 𝐯1=(-(3)/(5), (4)/(5), 0),...

Verify that the vectors $$ \mathbf{v}_{1}=\left(-\frac{3}{5}, \frac{4}{5}, 0\right), \mathbf{v}_{2}=\left(\frac{4}{5}, \frac{3}{5}, 0\right), \mathbf{v}_{3}=(0,0,1) $$ form an orthonormal basis for $R^{3}$ with respect to the Euclidean inner product, and then use Theorem $6.3 .2(b)$ to express the vector $\mathbf{u}=(1,-2,2)$ as a linear combination of $\mathbf{v}_{1}$ $\mathbf{v}_{2},$ and $\mathbf{v}_{3}$

Verify that the vectors
$$
\mathbf{v}_{1}=\left(-\frac{3}{5}, \frac{4}{5}, 0\right), \mathbf{v}_{2}=\left(\frac{4}{5}, \frac{3}{5}, 0\right), \mathbf{v}_{3}=(0,0,1)
$$
form an orthonormal basis for $R^{3}$ with respect to the Euclidean inner product, and then use Theorem $6.3 .2(b)$ to express the vector $\mathbf{u}=(1,-2,2)$ as a linear combination of $\mathbf{v}_{1}$ $\mathbf{v}_{2},$ and $\mathbf{v}_{3}$

Key Concepts

Orthonormal Basis

An orthonormal basis in a vector space is a set of vectors that are both orthogonal (each pair of distinct vectors has an inner product of zero) and normalized (each vector has a length or norm equal to one). This property simplifies many computations in linear algebra, particularly when expressing vectors as a sum of other vectors, since the coefficients in the linear combination can be directly obtained through inner products.

Euclidean Inner Product

The Euclidean inner product, commonly known as the dot product, is a way of multiplying two vectors to yield a scalar. This product measures the cosine of the angle between the vectors and, by doing so, helps to determine orthogonality. In the context of verifying an orthonormal basis, it is used to check that the inner product of different basis vectors is zero and that each vector's inner product with itself is one.

Linear Combination

A linear combination of vectors involves summing multiple vectors each multiplied by a scalar coefficient. In a vector space, any vector can be uniquely expressed as a linear combination of basis vectors. When the basis is orthonormal, these coefficients can be easily computed by taking the inner product of the vector with each basis vector, which provides a simple and direct method of decomposition.

Coefficient Calculation via Projections (Theorem 6.3.2(b))

The theorem referred to in the problem typically states that if an orthonormal basis is given, then any vector in the space can be expressed as a sum of projections onto each of the basis vectors. The coefficients in this linear combination are simply the inner products of the vector with each basis vector. This result is a powerful tool in vector space theory as it simplifies the process of decomposing any vector into its component parts.

Transcript

00:01 Okay, for those two three vectors, let's first check their orthogonality between each other.

00:08 I mean, v1 is equal to v2 is equal to 4 over 5, 3 over 5 is 0, v3 is equal to 0, 0, 1.

00:29 By the definition of the inner product, it's very easy for us to verify.

00:35 In the product between v1 and v3 is equal to v3 in the product between v2 and v3.

00:43 And those two inner products are equal to 0 because you can see the first two coordinates for v3 are all 0.

00:55 So that means we only need to check the inner product between v1 and v2.

01:02 By the definition, we know it is equal to minus 4 over 5 times 3 over 5.

01:18 It's easy for us to know it is equal to 0.

01:22 Okay, then to show they are orthonormal, we need to check the length of those three vectors.

01:31 And it's easy for us to show the length of the third one is equal to 1.

01:38 Now let's check the length of the second one.

01:43 And by the definition, it is equal to 4 over 5 squared plus 3 over 5 to the power 2.

01:51 And it's easy for us to know this dimension is equal to 1.

01:56 And we can do the same thing for the first one.

01:59 And we get it minus 3 over 5 to the power 2 plus 4 over 5 to the power 2.

02:06 So we can guess it's 1.

02:10 And notice those three vectors are vectors for r3...

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