Adding Vectors and Scaling by Scalars

Coordinates Determine the Operations

A vector in \(\mathbb{R}^n\) is an ordered list of \(n\) real coordinates, written as a column. Adding vectors and multiplying a vector by a scalar are basic ways to form new vectors from given ones. Both operations are performed on coordinates, but addition has an important requirement: the two vectors must have the same number of coordinates. A scalar multiple, by contrast, always has the same dimension as the vector being scaled.

For example, a vector in \(\mathbb{R}^3\) has three coordinates, so its first, second, and third coordinates can be matched with the corresponding coordinates of another vector in \(\mathbb{R}^3\). A vector in \(\mathbb{R}^2\) has no third coordinate to match with a vector in \(\mathbb{R}^3\). The dimensions are part of what specifies the vectors; they are not just a formatting choice.

Coordinate positions must be kept in order. In a sum, the first coordinate of one vector is added to the first coordinate of the other, the second to the second, and so on. For scalar multiplication, the same scalar is used on every coordinate. It is not a separate choice of scalar for each entry.

Why the Coordinate Rules Give the Right Results

Proof. Write \(\mathbf{u}=(u_1,\dots,u_n)^{\top}\) and \(\mathbf{v}=(v_1,\dots,v_n)^{\top}\). By the definition of the sum, its ordered list of coordinates is \((u_1+v_1,\dots,u_n+v_n)^{\top}\), so its \(i\)th coordinate is \(u_i+v_i\). By the definition of scalar multiplication, the coordinates of \(c\mathbf{v}\) are \((cv_1,\dots,cv_n)^{\top}\), so its \(i\)th coordinate is \(cv_i\). These descriptions apply to each \(i=1,\dots,n\), which proves both formulas. \(\square\)

Proof. The sum of two vectors in \(\mathbb{R}^n\) has one coordinate in each of the \(n\) positions, namely \(u_i+v_i\) in position \(i\). Each coordinate is real, so the result is in \(\mathbb{R}^n\). The scalar multiple has exactly the \(n\) coordinates \(cv_1,\dots,cv_n\), each real, so it too is in \(\mathbb{R}^n\). The coordinatewise definition pairs coordinate \(i\) of one input with coordinate \(i\) of the other for every position. This pairing requires both inputs to have the same number \(n\) of coordinates; if their dimensions differ, there is no matching coordinate in every position, so this definition does not give a sum. \(\square\)

These propositions make the practical procedure precise: determine the dimension first, then calculate entries in their original positions. For addition, the coordinates are paired position by position. For scaling, each coordinate is multiplied by the chosen scalar. Once calculated, the result should still have the expected number of entries.

Adding Vectors with Integer Coordinates

Negative coordinates are ordinary real numbers in these calculations. For instance, the third coordinate in the sum above is \(4+(-2)=2\). Parentheses help keep the coordinate calculation visible and prevent a sign from being lost.

Fractions and Scalar Multiples

When fractions appear, it is useful to write the coordinate operations before simplifying. This gives a separate check for each position: a mistake in one coordinate does not change the calculation in another, but it does change the resulting vector.

Different Dimensions Must Stay Distinct

A vector with two coordinates is not treated as a three-coordinate vector by silently adding another entry. Even when a missing entry might seem convenient, doing so changes the vector and is an extra operation, not part of vector addition as defined here. Always record the dimension before attempting a sum.

Common Mistakes

Padding a shorter vector to make addition work. A tempting but incorrect calculation is \((1,2)^{\top}+(3,4,5)^{\top}=(4,6,5)^{\top}\). This treats the first vector as though it were \((1,2,0)^{\top}\), although that is a different vector in \(\mathbb{R}^3\). The correction is that the displayed sum is not defined by coordinatewise addition: the first vector is in \(\mathbb{R}^2\) and the second is in \(\mathbb{R}^3\).

Scaling only some coordinates. Another incorrect calculation is \(3(1,2,-1)^{\top}=(3,2,-1)^{\top}\), where only the first coordinate was multiplied. The scalar rule applies to every coordinate, so the correct calculation is \(3(1,2,-1)^{\top}=(3\cdot1,3\cdot2,3\cdot(-1))^{\top}=(3,6,-3)^{\top}\). The coordinate count is unchanged, and each entry has been scaled.

Before finalizing a calculation, check two things: whether the inputs to a sum have the same dimension, and whether every coordinate required by the operation has been calculated. These checks catch many errors before any later algebra is needed.