Understanding Least Squares Solutions

This video tutorial explains how to determine least squares solutions for matrix equations where the matrix has orthogonal columns and no exact solution exists. It covers the setup of an example problem, verification of orthogonality, and calculation of the least squares solution using dot products. The tutorial also evaluates the closeness of the approximate solution to the original vector, providing insights into the accuracy of the least squares method.

What is the primary condition for using least squares solutions?

• Matrix A must be invertible.

• Matrix A must be square.

• Matrix A must have orthogonal columns.

• Vector b must be zero.

How is the least squares solution vector x hat denoted?

• As a polynomial

• As a vector

• As a scalar

• As a matrix

What is the significance of the dot products in the least squares solution formula?

• They verify the orthogonality of vector b.

• They determine the size of matrix A.

• They simplify the matrix multiplication.

• They are used to calculate the components of vector x hat.

What does a zero result from the dot product of two vectors indicate?

• The vectors are parallel.

• The vectors are orthogonal.

• The vectors are identical.

• The vectors are zero vectors.

What does the contradiction in row four of the augmented matrix indicate?

• There is a unique solution.

• There are infinite solutions.

• The solution is zero.

• There is no solution.

What is the result of vector b dotted with vector u sub one divided by vector u sub one dotted with itself?

• Zero

• Negative three halves

• Negative three

• Two

What is the least squares solution for the given matrix and vector?

• Vector negative three halves, negative three halves, two

• Vector one

• Vector two, two, two

• Vector zero

How close is the result of matrix A times vector x hat to vector b?

• One unit away

• Exactly the same

• Two units away

• Three units away

What does the magnitude of the difference between two vectors represent?

• The sum of the vectors

• The distance between their terminal points

• The product of the vectors

• The angle between the vectors

What is the result of multiplying matrix A by the least squares solution vector x hat?

• Vector zero

• Vector one half, one half, seven halves, seven halves

• Vector one, one, one, one

• Vector negative three halves, negative three halves, two