Find the change-of-basis matrix PC ←B from the given ordered basis B to the given ordered basis

step 1 we need to find the change of basis matrix from B to C. To do that first, we write the first vec

Find the change-of-basis matrix PC ←B from the given ordered...

Find the change-of-basis matrix $P_{C \leftarrow B}$ from the given ordered basis $B$ to the given ordered basis $C$ of the vector space $V.$
$$\begin{array}{l}V=\mathbb{R}^{3} ; B=\{(-7,4,4),(4,2,-1),(-7,5,0)\} \\C=\{(1,1,0),(0,1,1),(3,-1,-1)\} \end{array}.$$

Key Concepts

Change-of-Basis Matrix

A change-of-basis matrix, or transition matrix, is a matrix that converts the coordinate representation of any vector from one basis to another. It is constructed by expressing the vectors of the original basis in terms of the new basis, and this matrix facilitates the transformation of coordinates between different ordered bases within the same vector space.

Coordinate Representation

Coordinate representation is the expression of a vector as a unique linear combination of the basis vectors of a vector space. The coefficients or coordinates in this linear combination provide a convenient way to manipulate and analyze vectors, particularly when changing from one basis to another using a change-of-basis matrix.

Vector Space

A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars in a way that satisfies specific axioms such as associativity, commutativity, the existence of a zero vector, and distributivity. Vector spaces provide the framework for linear algebra and allow for the abstract study of Euclidean spaces and beyond.

Basis and Ordered Basis

A basis of a vector space is a set of linearly independent vectors that spans the entire space, meaning every vector in the space can be uniquely represented as a linear combination of the basis vectors. When the basis is ordered, the sequence in which the basis vectors are listed is significant; this order determines the coordinates of vectors when expressed relative to that basis.

Transcript

00:01 We need to find the change of basis matrix from b to c.

00:07 To do that first, we write the first vector of order basis b, which is negative 7, 4, and 4.

00:16 And then we write equal to, so we need to find a vector with three components, c1, c2, c3, that when it's multiplied by the vectors of ordered basis c, c, gives us the, or a vector i've written.

00:33 So what we do, we write c1 multiplied by the first vector, 1, 1, and 0, plus c2 is 0, 1, and 1.

00:45 And then c3, multiply, is 3, negative 1, negative 1.

00:52 Then here you have x, y, z, and x is equal to c1, multiple, y, x of the first vector, c2 multiply by the x of the second vector and c3 multiply by the x of the third vector and similarly for y and z so i write the system of equation which is c1 plus c3 c3 equals to negative 7 c1 plus c2 negative minus c3 is equal to 4 and then c2 minus c3 is equal to 4 and then c2 minus c3 is equal to 4 just we can rewrite this one c2 is equals to 4 plus c3 and replace it here then we have this is now for which we can solve it easily and we get c1 c2 and c3 equals to and 0 5 thid and negative 7th so the first vector relative to is c is 0 5.

02:18 So first we found the first column of the matrix we are looking for.

02:25 So we do the same for the second vector of the order basis b.

02:29 So i write it here.

02:31 The second vector is 4, 2, and negative 1.

02:37 Similarly, we write c1, 2, 1 ,0, plus c2 multiplied by 0, 1, and 1, plus c3, multiply, minus 3, negative 1, and 1.

02:59 I write the system of equations here.

03:02 We have c1 plus 3, c2, 3, equals to 4, c1, plus c2, minus c3 is supposed to 2, and c2.

03:20 Plus negative c3 equals to negative 1.

03:27 We can simplify here, say c2 equals to negative 1 plus c3, and then replace and solve for c1, c2, and c3.

03:39 I find the vector here, the second vector relative to c, which is going to be 3, negative 2 3, and 2 .3, and one third...

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