Given 𝐚=(a, b, c)^T ∈ℝ^3, prove that the cross product map L𝐚[𝐯]=𝐚 ×𝐯, as defined in (4.2), is l

Given 𝐚=(a, b, c)^T ∈ℝ^3, prove that the cross product map...

Key Concepts

Skew-Symmetric Matrices

A skew-symmetric matrix is a square matrix whose transpose is equal to its negative. In three-dimensional space, the cross product with a fixed vector can be represented by a skew-symmetric matrix. This is a key concept because the matrix that represents the cross product operator is constructed such that its structure inherently captures the antisymmetric property of the cross product, leading to the conclusion that the map is linear.

Linear Transformation

A linear transformation is a function between vector spaces that preserves the operations of vector addition and scalar multiplication. This means that when two vectors are added or when a vector is multiplied by a scalar, the transformation of the result is equal to the sum or scalar multiple of the individual transformations, respectively. Demonstrating that the cross product operator is linear involves verifying these properties, which is a central idea when representing such operations using matrices.

Matrix Representation of a Linear Map

Every linear map between finite-dimensional vector spaces can be represented as a matrix once a basis is chosen for each space. This connection is crucial because it allows for the abstract operation of the linear map (such as the cross product with a fixed vector) to be expressed concretely in terms of matrix multiplication. Finding the matrix representation involves determining the images of the basis vectors under the transformation and forming a matrix where the columns are these images.

Transcript

00:01 Hello everybody.

00:02 In this video, i'm going to be showing you how to solve exercise 68 in chapter 13, section 4 of calculus early transcendental.

00:10 Now, in this problem, they want us to prove that a vector cross -width itself always produces the zero vector, and they want us to prove this in three different ways.

00:21 The first method they want us to use is the definition of the cross -product itself.

00:26 As per this definition, the vector u cross -v between two vectors u and v is a vector that has a magnitude, which is equal to the magnitude of u times the magnitude of v, times the sign of the angle between u and v.

00:45 So this means that u cross with view is a vector that is equal to the magnitude of u times the magnitude of u squared, times the sign of zero, as, of course, the angle of any vector with itself is zero.

01:09 They are not separated by an angle because they're the same vector.

01:12 But the sign of zero is equal to zero, which makes this whole quantity equal to zero.

01:18 And now the only vector that has a magnitude of zero is the zero vector itself.

01:25 So we can conclude that root u cross u equals the zero vector.

01:34 And so that's how you prove this statement using the definition of the cross product.

01:38 But now let's go on to using the determinant method.

01:42 As per theorem 13 .6, we can express u as a vector with components, u1, u2, and u3.

01:54 And this means that u cross u is equal to the determinant of a 3 by 3 matrix with the standard unit vectors in the top row, the components of u in the second row, as well as in the third row.

02:14 And now if we expand this determinant, we have that the first term is equal to i times the determinant.

02:19 Of this 2x 2 by 2 sub matrix, u2, u3, u2, u3, minus a similar quantity for j, j times this the sub -matrix formed by this bottomless sub -colum and bottom right column, which we take the determinant of u1, u3, u3 again...

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