Understanding Vector Span and Linear Combinations

The video tutorial explains how to determine if a vector is in the span of two other vectors by expressing it as a linear combination. It involves setting up a vector equation, converting it into a system of equations, and solving it using augmented matrices. The process is verified by checking the solution, confirming that the vector is indeed a linear combination of the given vectors.

What must be true for vector w to be in the span of vectors u and v?

• Vector w must be perpendicular to vectors u and v.

• Vector w must be a linear combination of vectors u and v.

• Vector w must have the same magnitude as vectors u and v.

• Vector w must be parallel to vectors u and v.

What does it mean for vector w to be a linear combination of vectors u and v?

• Vector w is the sum of vectors u and v.

• Vector w is orthogonal to vectors u and v.

• Vector w is a scalar multiple of vector u.

• Vector w can be expressed as a sum of scalar multiples of vectors u and v.

How are vectors expressed in the vector equation?

• As diagonal matrices

• As scalar values

• As column matrices

• As row matrices

What is the result of performing scalar multiplication and addition on the left side of the vector equation?

• A 3x3 matrix

• A 1x2 row matrix

• A 2x2 matrix

• A 2x1 column matrix

What is the first step in solving the system of equations derived from the vector equation?

• Writing an augmented matrix

• Performing matrix multiplication

• Finding the determinant

• Calculating the inverse

What is the purpose of transforming the augmented matrix to reduced row echelon form?

• To find the inverse of the matrix

• To calculate the determinant

• To determine the rank of the matrix

• To solve for the scalars c1 and c2

What is the least common multiple used to eliminate variables in the augmented matrix?

• 4

• 5

• 6

• 7

What operation is performed to make the leading entries of the matrix positive one?

• Adding a row to itself

• Dividing a row by its leading entry

• Multiplying a row by -1

• Swapping two rows

What are the values of c1 and c2 that verify vector w is in the span of vectors u and v?

• c1 = 5, c2 = 2

• c1 = 7, c2 = -1

• c1 = 1, c2 = 0

• c1 = 3, c2 = 4

How can we verify that vector w is in the span of vectors u and v?

• By ensuring vector w has the same magnitude as vectors u and v

• By checking if vector w is orthogonal to vectors u and v

• By expressing vector w as a linear combination of vectors u and v

• By calculating the dot product of vector w with vectors u and v