() S = and v = 3 1. Find the cosine of the angle between the first two vectors in S. 2. Find the length of the third vector in S. 3. Find the matrix of the standard inner product on W (obtained by restriction of the dot product on R') with respect to the basis S. 4. Orthonormalize S (Gram-Schmidt). 5. Find the projection of v onto W.

() S = and v = 3 1. Find the cosine of the angle between the first two vectors in S. 2. Find the length of the third vector in S. 3. Find the matrix of the standard inner product on W (obtained by restriction of the dot product on R') with respect to the basis S. 4. Orthonormalize S (Gram-Schmidt). 5. Find the projection of v onto W.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 18CM

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

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Consider the subspace W of R⁴ spanned by the set S and a vector v in the attachment, and please solve (1) - (3). Thank you!

()
S =
and
v =
3
1. Find the cosine of the angle between the first two vectors in S.
2. Find the length of the third vector in S.
3. Find the matrix of the standard inner product on W (obtained by
restriction of the dot product on R') with respect to the basis S.
4. Orthonormalize S (Gram-Schmidt).
5. Find the projection of v onto W.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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