Difference Between a Row & Column Vector Video

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  • 0:04 Vectors
  • 0:54 Row and Column Vectors
  • 1:58 Operations
  • 3:41 Lesson Summary
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Lesson Transcript
Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

Learn how row and column vectors differ by viewing them through matrix notation. Then explore some unique interactions between row and column vectors by seeing how a couple types of matrix operations work on them.


In an introductory linear algebra course, you'll spend a lot of time working with vectors. You've probably already learned that a vector is different from a scalar in that it has both magnitude and direction, and you've seen them written out as an ordered list of elements. That's basically what it is, an ordered list of elements, and differs from a scalar by having both magnitude and direction.

Once you understood what a vector is, you then moved on to learning some of the basic vector operations. Examples of these include vector addition, subtraction, scalar multiplication, dot product, and cross product.

column and row vectors

While you've been working with vectors, you may have noticed that they tend to be written in one of two different ways. Vectors can be written vertically or horizontally. We call these column and row vectors respectively. In this lesson, you're going to learn what differentiates a column vector from a row vector.

Row and Column Vectors

In order to understand what makes column and row vectors different from each other, we actually need to start by looking at matrices, not vectors. A matrix is a rectangular array of elements. We normally categorize a matrix by its dimensions, which are written as the number or rows in the matrix multiplied by the number of columns in it. For example, a 2x2 matrix has 2 rows and 2 columns, a 3x4 matrix has 3 rows and 4 columns, and an nxm matrix has n rows and m columns.

These example matrices show how we determine the dimensions of a matrix.
matrix dimensions

Understanding how matrices are categorized by dimension is the trick to seeing the difference between column and row vectors. Vectors can be viewed as a special type of matrix, where one of their two dimensions is always equal to 1. Depending on which dimension is set to 1, you'll get either a column or a row vector. A column vector is an nx1 matrix because it always has 1 column and some number of rows. A row vector is a 1xn matrix, as it has 1 row and some number of columns. This is the major difference between a column and a row vector.

The dimensions of row and column vectors are determined in the same way as the dimensions of a matrix.
row and column vector dimensions


Since column and row vectors can be viewed as matrices, it also makes sense that matrix operations can also be performed on them. Let's go over a couple matrix operations that give interesting results when performed on column and row vectors.

The first operation we will go over is matrix transposition. When you take the transpose of a matrix, you switch the rows and columns inside it. All the rows of the matrix become columns and vice versa. Since vectors only have 1 row or 1 column, this operation actually switches the type of vector they are. A column vector becomes a row vector and a row vector becomes a column vector.

These examples show how you take the transpose of a matrix, a row vector, and a column vector.
matrix and vector transposition

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