Computing Linear Combinations from Coefficients

A Reliable Way to Evaluate a Linear Combination

A linear combination is a weighted sum of vectors: each vector is multiplied by its assigned scalar, and the resulting vectors are added. When the vectors and coefficients are given, the task is to carry out those operations accurately. A dependable calculation keeps the coefficient attached to the correct vector, scales that entire vector, and only then adds the results coordinate by coordinate.

Throughout, vectors are columns in \(\mathbb{R}^n\), and the coefficients are real numbers. For example, in \(2\mathbf{u}-3\mathbf{v}\), the coefficient \(2\) belongs to \(\mathbf{u}\), while the coefficient \(-3\) belongs to \(\mathbf{v}\). The minus sign is part of the second coefficient. It is often helpful to rewrite the expression as \(2\mathbf{u}+(-3)\mathbf{v}\), so that every term visibly consists of one coefficient and its vector.

Vector addition and scalar multiplication in \(\mathbb{R}^n\) are performed coordinatewise. Thus, to multiply a vector by a scalar, multiply every coordinate by that scalar. Then add the scaled vectors: add their first coordinates to get the first coordinate of the answer, their second coordinates to get the second coordinate, and continue through the list. Keep the coordinates in their original order throughout; the first coordinate of one vector must be added to the first coordinate of every other vector, not to a different position.

The Calculation in Three Stages

Keeping the stages separate helps locate mistakes. If the scaled vectors are correct but the final answer is not, the error occurred during addition. If a scaled vector is wrong, there is no need to redo the whole sum before correcting it. For exact arithmetic, retain integers and fractions instead of converting them to decimals. This makes signs and equalities easier to inspect.

The check can be done using the original expression without repeating every vector addition. In each coordinate, take that coordinate from each original vector, multiply by its coefficient, and add. This is a checking procedure for a particular calculation; it does not change the order of the main method, which is to scale the vectors and then add them.

Zero and Negative Coefficients

Some coefficients need special attention. A zero coefficient contributes a zero vector, regardless of the entries in the vector it multiplies. It is still useful to keep the term visible until the other terms have been matched correctly. A negative coefficient multiplies every coordinate, not just the coordinates that were positive. For instance, multiplying \(\begin{bmatrix}0\\2\end{bmatrix}\) by \(-1\) gives \(\begin{bmatrix}0\\-2\end{bmatrix}\): the zero coordinate stays zero.

Proof. The distributive law for scalar multiplication gives \((0+0)\mathbf{v}=0\mathbf{v}+0\mathbf{v}\). The left side equals \(0\mathbf{v}\), so this equation says \(0\mathbf{v}=0\mathbf{v}+0\mathbf{v}\). Add the additive inverse of \(0\mathbf{v}\) to both sides. The left side becomes \(\mathbf{0}\), and the right side becomes \(0\mathbf{v}\). Hence \(0\mathbf{v}=\mathbf{0}\). Adding a zero vector to any vector leaves it unchanged, so a zero-coefficient term does not affect the sum. \(\square\)

Combining Terms That Use the Same Vector

Sometimes the same vector appears more than once. The terms can be combined by adding their coefficients, which may shorten the arithmetic. The coefficient addition must preserve the signs: for example, \(4\mathbf{u}-\mathbf{u}\) has combined coefficient \(4+(-1)=3\), so it equals \(3\mathbf{u}\).

Proof. The distributive law for scalar multiplication states that \((a+b)\mathbf{u}=a\mathbf{u}+b\mathbf{u}\). Since scalar addition is commutative, the right side can be written as \(a\mathbf{u}+b\mathbf{u}\) in either order. Therefore the two expressions are equal. This argument also applies when either coefficient is zero or negative, because the distributive law holds for all real scalars. \(\square\)

Combining repeated terms is optional. It can make a calculation shorter, but it adds a step at which coefficients can be added incorrectly. When the expression is short, scaling each term as written may be safer. When combining terms, first identify which vectors are exactly the same, then add only their coefficients, including their signs.

Common Mistakes and How to Correct Them

Mistake: changing the sign of a zero coordinate. A negative scalar reverses the sign of each nonzero coordinate, but zero has no positive or negative sign to reverse. A wrong calculation would be \((-1)\begin{bmatrix}0\\2\end{bmatrix}=\begin{bmatrix}-1\\-2\end{bmatrix}\). The first entry is wrong because \((-1)(0)=0\). The correction is

Mistake: applying the coefficient to only one coordinate. For example, \(-2\begin{bmatrix}1\\3\end{bmatrix}\) is not \(\begin{bmatrix}-2\\3\end{bmatrix}\). The coefficient multiplies both entries, giving \(\begin{bmatrix}-2\\-6\end{bmatrix}\). Before adding vectors, check every entry of each scaled vector.

Mistake: adding unlike coordinates. If two vectors have two coordinates, add the first entries together and the second entries together. For \(\begin{bmatrix}2\\-1\end{bmatrix}+\begin{bmatrix}5\\4\end{bmatrix}\), the result is \(\begin{bmatrix}2+5\\-1+4\end{bmatrix}=\begin{bmatrix}7\\3\end{bmatrix}\), not a vector formed by adding entries across different positions.

Mistake: dropping a minus sign before scaling. The expression \(2\mathbf{u}-3\mathbf{v}\) has coefficients \(2\) and \(-3\), not \(2\) and \(3\). Writing \((-3)\mathbf{v}\) explicitly before multiplying helps prevent this error. After the computation, check each output coordinate against the original coefficients, as in the examples.

These checks are especially useful when a coordinate contains more than one negative number. A negative coefficient times a negative coordinate gives a positive entry, while a negative coefficient times a positive coordinate gives a negative entry. For instance, \((-3)(-2)=6\), whereas \((-3)(2)=-6\). The sign of the product depends on both signs, so it is worth writing each small multiplication before summing.