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Linear Algebra: Vector Operations and Linear Combinations

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Vector Addition and Scalar Multiplication

Given two vectors, v=2i−5j\mathbf{v} = 2\mathbf{i} - 5\mathbf{j} and w=−3i+7j\mathbf{w} = -3\mathbf{i} + 7\mathbf{j}, perform the following operations:

(A) v+w\mathbf{v} + \mathbf{w}

(B) v−w\mathbf{v} - \mathbf{w}

(C) 2v+3w2\mathbf{v} + 3\mathbf{w}

(D) 4v−5w4\mathbf{v} - 5\mathbf{w}

Vector Operations and Linear CombinationsEasyVideo
Addition of Two Vectors

Given two vectors X=[12]X = \begin{bmatrix} 1 \\ 2 \end{bmatrix} and Y=[−46]Y = \begin{bmatrix} -4 \\ 6 \end{bmatrix}, compute X+YX + Y.

Vector Operations and Linear CombinationsEasyVideo
Subtracting Two Vectors

Subtract the vector Y=[−46]Y = \begin{bmatrix} -4 \\ 6 \end{bmatrix} from X=[12]X = \begin{bmatrix} 1 \\ 2 \end{bmatrix}. Compute X−YX - Y.

Vector Operations and Linear CombinationsEasyVideo
Calculating Linear Combination of Vectors

Calculate the linear combination 3X+2Y3X + 2Y for the vectors X=[12]X = \begin{bmatrix} 1 \\ 2 \end{bmatrix} and Y=[−46]Y = \begin{bmatrix} -4 \\ 6 \end{bmatrix}.

Vector Operations and Linear CombinationsEasyVideo
Vector Addition

Given vectors A=[124]A = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix} and B=[027]B = \begin{bmatrix} 0 \\ 2 \\ 7 \end{bmatrix}, compute A+BA + B.

Vector Operations and Linear CombinationsEasyVideo
Vector Subtraction

For vectors A=[124]A = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix} and B=[027]B = \begin{bmatrix} 0 \\ 2 \\ 7 \end{bmatrix}, compute B−AB - A.

Vector Operations and Linear CombinationsEasyVideo
Calculate Expression for Vector Sum

Calculate the expression B+3AB + 3A for vectors A=[124]A = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix} and B=[027]B = \begin{bmatrix} 0 \\ 2 \\ 7 \end{bmatrix}.

Vector Operations and Linear CombinationsEasyVideo