Question

Let b1 = [-2,1] and b2 = [-1,3]. Let B be the basisi for R²

What is the transition matrix PB from basis B to standardbasis for R²
Let x = [5,10]. Find [x]B, the coordinate vector of x in the basis B.
Let c1 = [1,2] and c2 = [4,3] , and C the basis {c1,c2}.

Answer 1

The transition matrix PB from basis B to the standard basis for R² is achieved by arranging the basis vectors as columns in a matrix. Coordinate vector [x]B can be found through matrix algebra. The outcomes of vector cross and dot products inform the relationship between the vectors, i.e., parallel or orthogonal.

Step-by-step explanation:

Transition Matrix and Coordinate Vector in Different Bases

To find the transition matrix PB from basis B to the standard basis for ℝ², we would place the basis vectors b1 and b2 as columns in a matrix:

PB = [-2, -1;1, 3]

This matrix directly translates coordinates from the B basis to the standard basis for 2-dimensional real space. To find the coordinate vector of x in basis B, which is given by [5,10] in the standard basis, we solve PB [x]B = x for [x]B. We can use matrix algebra to solve for [x]B.

In the new basis C, comprised of vectors c1 and c2, we can similarly construct a transition matrix and find the coordinates of any vector in that basis.

For vector products, when considering the cross product, if it vanishes, it implies that the vectors are parallel, while a vanishing dot product implies that the vectors are orthogonal to each other. The dot product of a vector with the cross product that this vector has with another vector will always result in 0 because a cross product is perpendicular to both original vectors, thus it's orthogonal to the first vector when taking the dot product.